text:
\section{Introduction}

Compactness is a crucial property of topological space which play an
important role for the development of topological entropy of a dynamical
system. Adler et al. \cite{adler} introduced the concept of topological
entropy of continuous mapping for a compact space of dynamical system in
1965. Subsequently, Bowen \cite{bowen} generalized the concept of
topological entropy due to Adler et al. \cite{adler} which is
metric-dependent and hold for noncompact spaces, see also \cite{zabidin} for
the detail explanation of these two notions. Next, Liu et al. \cite{liu}
generalized the notion of Adler et al. where separation axioms not
necessarily required which intiate the concept of topological entropy for
arbitrary topological spaces. In 2015, Afsan \cite{uzzal} defined
topological $H$-entropy for topological $H$-dynamical system based on $H
-set and $H$-closed topological space.

In a recent paper, Gulamsarwar and Salleh \cite{gulam} presented a new
generalized notion of topological entropy on nearly compact space by using
R $-map \cite{carnahan}, namely topological nearly entropy. Recall that
every compact space is nearly compact but not the converse. Thus, the
concept of topological nearly entropy for topological $R$-dynamical system
was initiated in accordance with several fundamental properties of Adler et
al. \cite{adler} and Liu et al. \cite{liu}. Until now, many researchers are
interested with the concept of topological entropy as we can see more of
their works in \cite{canovas} and \cite{goodwyn}, etc.

This paper explore more fundamental properties of topological nearly
entropy. Moreover this paper presenting the notion of topological nearly
entropy for $R$-map on nearly compact space and some its fundamental
properties are studied. These fundamental properties are further extension
of Gulamsarwar and Salleh \cite{gulam} topological nearly entropy.
Furthermore, we introduce $R$-space in order to study the effect of nearly
compactness on Hausdorff space to the topological nearly entropy. Finally,\
we show that $R$-map preserves nearly compact property and obtain several
properties of topological nearly entropy for product space.

In Section 3, we explore more properties and show the results obtained for
topological nearly entropy of the funtion $f$ and it inverse $f^{-1}$ if $f$
is bijective. The definition of topological nearly entropy on nearly compact
space and its fundamental properties are discussed in Section 4. We also
study topological nearly entropy on nearly compact and Hausdorff space which
has triggered the idea to introduce $R$-space. Consequently, we obtain
several properties of topological nearly entropy for product space which are
discusses in the Section 5.

\section{Preliminaries}

Throughout this paper $X$ is always mean arbitrary topological space unless
explicitly stated. We denote the closure and interior of $A\subseteq X$ by
\limfunc{cl}\left( A\right) $ and $\limfunc{int}\left( A\right) $,
respectively. A subset $A$ of $X$ is said to be regular open if and only if
A=\limfunc{int}\left( \limfunc{cl}\left( A\right) \right) $. By regular open
cover of $X$, we mean a cover of $X$ by regular open sets of $X$. The join
of two covers $\mathcal{U}$ and $\mathcal{V}$ are described as $\mathcal{U
\vee \mathcal{V}=\left\{ A\cap B:A\in \mathcal{U},B\in \mathcal{V}\right\} $
and a cover $\mathcal{V}$ is said to be refinement of a cover $\mathcal{U}$,
denoted as $\mathcal{U}\prec \mathcal{V}$, if for every element $V\in
\mathcal{V}$ there exists an element $U\in \mathcal{U}$ such that
V\subseteq U$. The complement of a set $A$ is denoted as $A^{c}$.

A subset $Y$ of $X$ is said to be nearly compact relative to $X$ if every
cover $\mathcal{U}=\{U_{\alpha }:\alpha \in \Delta \}$ of $X$ by open sets
of $X$ contains a finite subfamily $\mathcal{U}_{0}=\{U_{1},U_{2},\ldots
,U_{n}\}$ such that $Y\subseteq \bigcup_{i=0}^{n}\limfunc{int}\left(
\limfunc{cl}\left( U_{i}\right) \right) $, or equivalently, $Y$ is covered
by finitely regular open sets of $X$. If $Y=X$ is nearly compact relative to
$X$, the space $X$ is called nearly compact. In other words, the space $X$
is nearly compact if and only if every regular open cover of $X$ contains a
finite subcover, see \cite{singal}.

A function $f:X\rightarrow Y$ is called $R$-map \cite{carnahan} if the
inverse image of every regular open set in $Y$ is regular open in $X$. If
f:X\rightarrow X$ is an $R$-map, then the pair $\left( X,f\right) $ is
called topological $R$-dynamical system, and if $X$ is nearly compact,
\left( X,f\right) $ is called nearly compact $R$-dynamical system, see \cit
{gulam}. Now, we recall some definitions of topological nearly entropy and
its fundamental properties from \cite{gulam} as follows.

\begin{definition}
\cite{gulam} Let $\left( X,f\right) $ be a topological $R$-dynamical system,
$\mathcal{U}$ be a regular open cover of $X$ and $K$ be a nonempty nearly
compact relative to the space $X$ such that $f\left( K\right) \subseteq K$.
Let $N_{K}\left( \mathcal{U}\right) =\min \left\{ \limfunc{card}\left(
\mathcal{V}\right) :\mathcal{V}\text{ is a subcover of }\mathcal{U
,K\subseteq \dbigcup\limits_{V\in \mathcal{V}}V\right\} $. Since $K$ is a
nearly compact relative to $X$, $N_{K}\left( \mathcal{U}\right) $ is a
positive integer. Let $M_{K}\left( \mathcal{U}\right) =\log N_{K}\left(
\mathcal{U}\right) $.
\end{definition}

\begin{theorem}
\label{Th2.1}\cite{gulam} Let $\left( X,f\right) $ be a topological $R
-dynamical system. Let $\mathcal{U}$ and $\mathcal{V}$ be a regular open
covers of $X$, and $K$ be a nearly compact relative to $X$ such that
f\left( K\right) \subseteq K$. Then the following statements hold:

$\left( a\right) $ $M_{K}\left( \mathcal{U}\right) \geq 0$

$\left( b\right) $ $\mathcal{U}\prec \mathcal{V}$ implies $M_{K}\left(
\mathcal{U}\right) \leq M_{K}\left( \mathcal{V}\right) $

$\left( c\right) $ $M_{K}\left( \mathcal{U}\vee \mathcal{V}\right) \leq
M_{K}\left( \mathcal{U}\right) +M_{K}\left( \mathcal{V}\right) $

$\left( d\right) $ $M_{K}\left( f^{-1}\left( \mathcal{U}\right) \right) \leq
M_{K}\left( \mathcal{U}\right) $. When $f\left( K\right) =K$, the equality
holds.
\end{theorem}

Let $f:X\rightarrow X$ be an $R$-map and let $\left( X,f\right) $ be a
topological $R$-dynamical system. Denote by $H\left( X,f\right) $ the set of
all $f$-invariant nonempty nearly compact relative to $X$, i.e.,
\begin{equation*}
\left\{ K\in P\left( X\right) \setminus \left\{ \emptyset \right\} :\text{ }
\text{ is nearly compact relative to }X\text{ such that }f(K)\subseteq
K\right\} .
\end{equation*
If $X$ is nearly compact, it follows from $f\left( X\right) \subseteq X$
that $H\left( X,f\right) \neq \emptyset $. However, when $X$ is not nearly
compact, $H\left( X,f\right) $ could be empty.

\begin{definition}
\cite{gulam}\label{def_N1} Let $\left( X,f\right) $ be a topological $R
-dynamical system. $H\left( X,f\right) $ be the set $\left\{ K\in P\left(
X\right) \setminus \left\{ \emptyset \right\} :K\text{ is nearly compact
relative to }X\text{ such that }f\left( K\right) \subseteq K\right\} $ and
\mathcal{U}$ be a regular open cover of $X$. Then for $K\in H\left(
X,f\right) $
\begin{equation*}
Ent_{N}\left( f,\mathcal{U},K\right) =\lim_{n\rightarrow \infty }\frac{1}{n
M_{K}\left( \bigvee_{i=0}^{n-1}f^{-i}\left( \mathcal{U}\right) \right)
\end{equation*
is called the topological nearly entropy of $f$ on $K$ relative to $\mathcal
U}$ and
\begin{equation*}
Ent_{N}\left( f,K\right) =\sup_{\mathcal{U}}\left\{ Ent_{N}\left( f,\mathcal
U},K\right) :\mathcal{U}\ \text{is a regular open cover of}\ X\right\}
\end{equation*
is called topological nearly entropy of $f$ on $K$.
\end{definition}

\begin{theorem}
\cite{gulam}\label{Th 2.2} Let $\left( X,f\right) $ be a topological $R
-dynamical system, $K_{1},K_{2}\in H\left( X,f\right) $ with $K_{1}\subseteq
K_{2}$ and $\mathcal{U}$ be a regular open cover of $X$. Then

$\left( a\right) $ $Ent_{N}\left( f,\mathcal{U},K_{1}\right) \leq
Ent_{N}\left( f,\mathcal{U},K_{2}\right) $ and

$\left( b\right) $ $Ent_{N}\left( f,K_{1}\right) \leq Ent_{N}\left(
f,K_{2}\right) $.
\end{theorem}

\begin{definition}
\cite{gulam}\label{def_N2} Let $\left( X,f\right) $ be a topological $R
-dynamical system. The
\begin{equation*}
Ent_{N}\left( f\right) =\sup_{K}\left\{ Ent_{N}\left( f,K\right) :K\in
H\left( X,f\right) \right\}
\end{equation*
is called topological nearly entropy of $f$. For the special case $H\left(
X,f\right) =\emptyset ,Ent_{N}\left( f\right) =0$ is to be taken.
\end{definition}

\section{Further Properties of Topological Nearly Entropy}

Several fundamental properties of topological nearly entropy has been proved
in our previous paper (see \cite{gulam}). Now we shall prove another further
result about topological nearly entropy. The topological nearly entropy of a
map $f$ is coincide with topological nearly entropy of it inverse $f^{-1}$
whenever $f,f^{-1}$ are bijections and $R$-maps as follows.

\begin{theorem}
Let $\left( X,f\right) $ be a topological $R$-dynamical system and the
function $f:X\rightarrow X$ is bijective. Assume that $f^{-1}:X\rightarrow X$
is $R$-map, then if $K\in H\left( X,f\right) $ and $f,f^{-1}:K\rightarrow K$
are bijections and $R$-maps, then $Ent_{N}\left( f\right) =Ent_{N}\left(
f^{-1}\right) $.
\end{theorem}

\begin{proof}
If $H\left( X,f\right) =\emptyset $, then $Ent_{N}\left( f\right) =0$ and
Ent_{N}\left( f\right) \leq Ent_{N}\left( f^{-1}\right) $. So, let $H\left(
X,f\right) \neq \emptyset $. Now consider $K\in H\left( X,f\right) $, from
the asumption $f,f^{-1}:K\rightarrow K$ are bijections and $R$-maps, we have
$K\in H\left( X,f^{-1}\right) $ so that $H\left( X,f\right) \subseteq
H\left( X,f^{-1}\right) $. Let $\mathcal{U}$ be any regular open cover of $X$
and $K\in H\left( X,f\right) $. Since $f:K\rightarrow K$ is bijective and $R
-map, then $f\left( K\right) =K$. By Theorem 2.1(d) in \cite{gulam},
M_{K}\left( f^{-1}\left( \mathcal{U}\right) \right) =M_{K}\left( \mathcal{U
\right) $. So
\begin{eqnarray*}
Ent_{N}\left( f^{-1},\mathcal{U},K\right) &=&\lim_{n\rightarrow \infty
\frac{1}{n}M_{K}\left( \bigvee_{i=0}^{n-1}f^{i}\left( \mathcal{U}\right)
\right) \\
&=&\lim_{n\rightarrow \infty }\frac{1}{n}M_{K}\left( f^{-\left( n-1\right)
}\left( \bigvee_{i=0}^{n-1}f^{i}\left( \mathcal{U}\right) \right) \right) \\
&=&\lim_{n\rightarrow \infty }\frac{1}{n}M_{K}\left(
\bigvee_{i=0}^{n-1}f^{-i}\left( \mathcal{U}\right) \right) \\
&=&Ent_{N}\left( f,\mathcal{U},K\right) .
\end{eqnarray*
Then, we have $Ent_{N}\left( f\right) \leq Ent_{N}\left( f^{-1}\right) $ by
Definitions \ref{def_N1} and \ref{def_N2}.

By interchange the role of $f$ and $f^{-1}$ in the preceding argument, we
will have $Ent_{N}\left( f^{-1}\right) \leq Ent_{N}\left( \left(
f^{-1}\right) ^{-1}\right) =Ent_{N}\left( f\right) $. Thus, $Ent_{N}\left(
f\right) =Ent_{N}\left( f^{-1}\right) $.
\end{proof}

\section{Topological Nearly Entropy on Nearly Compact Spaces}

In this section, we define topological nearly entropy on nearly compact
space by using $R$-map. Moreover, we show that this definition is equivalent
to the previous definition in, see Definition \ref{def_N2} whenever $X$ is
nearly compact.

\begin{definition}
\label{topo n} Let $\left( X,f\right) $ be a nearly compact $R$-dynamical
system and $\mathcal{U}$ be a regular open cover of $X$. Le
\begin{equation*}
N_{n}\left( \mathcal{U}\right) =\min \left\{ \limfunc{card}\left( \mathcal{V
\right) :\mathcal{V}\text{ is a subcover of }\mathcal{U},X=\bigcup_{V\in
\mathcal{V}}V\right\} .
\end{equation*
Since $X$ is nearly compact, $N_{n}\left( \mathcal{U}\right) $ is a positive
integer. Let $M_{n}\left( \mathcal{U}\right) =\log N_{n}\left( \mathcal{U
\right) .$
\end{definition}

\begin{theorem}
\label{thm_n2.1} Let $\left( X,f\right) $ be a nearly compact $R$-dynamical
system and $\mathcal{U}$ and $\mathcal{V}$ be a regular open covers of $X$.
Then the following statements hold:

\begin{enumerate}
\item[$\left( a\right) $] $M_{n}\left( \mathcal{U}\right) \geq 0$.

\item[$\left( b\right) $] $\mathcal{U}\prec \mathcal{V}$ implies
M_{n}\left( \mathcal{U}\right) \leq M_{n}\left( \mathcal{V}\right) $.

\item[$\left( c\right) $] $M_{n}\left( \mathcal{U}\vee \mathcal{V}\right)
\leq M_{n}\left( \mathcal{U}\right) +M_{n}\left( \mathcal{V}\right) $.

\item[$\left( d\right) $] $M_{n}\left( f^{-1}\left( \mathcal{U}\right)
\right) \leq M_{n}\left( \mathcal{U}\right) $. The equality holds when $f$
is onto.
\end{enumerate}
\end{theorem}

\begin{proof}
The proofs are almost analogous to the proof of Theorem \ref{Th2.1} (see
\cite{gulam}) by replacing the role of $K$ as $X$ and slight modification of
the proof, one can easily establish this theorem.
\end{proof}

\begin{theorem}
\label{lim} Let $\left( X,f\right) $ be a nearly compact $R$-dynamical
system and $\mathcal{U}$ be a regular open cover of $X$. Then
\begin{equation*}
\lim_{m\rightarrow \infty }\frac{1}{m}M_{n}\left(
\bigvee_{i=0}^{m-1}f^{-i}\left( \mathcal{U}\right) \right)
\end{equation*
exists.
\end{theorem}

\begin{proof}
The proof are almost analogous to the proof af Theorem 2.3 in \cite{gulam}
by replacing the role of $K$ as $X$ and slight modification of the proof,
one can easily establish this theorem.
\end{proof}

\begin{definition}
\label{def topo N} Let $\left( X,f\right) $ be a nearly compact $R
-dynamical system and $\mathcal{U}$ be a regular open cover of $X$. Then
\begin{equation*}
Ent_{n}\left( f,\mathcal{U}\right) =\lim_{m\rightarrow \infty }\frac{1}{m
M_{n}\left( \bigvee_{i=0}^{m-1}f^{-i}\left( \mathcal{U}\right) \right)
\end{equation*
is called topological nearly entropy of $f$ relative to $\mathcal{U}$ and
\begin{equation*}
Ent_{n}\left( f\right) =\sup_{\mathcal{U}}\left\{ Ent_{n}\left( f,\mathcal{U
\right) :\mathcal{U}\mathrm{\ }\text{is~a regular~open~of}~X\right\}
\end{equation*
is called the topological nearly entropy of $f$.
\end{definition}

In the next theorem, we show the relationship between this new notion with
the previous Definition \ref{def_N1}.

\begin{theorem}
\label{n=N} Let $\left( X,f\right) $ be a topological $R$-dynamical system,
K\in H\left( X,f\right) $ and $\mathcal{U}$ be any regular open cover of $X
. Then for the subspace $K$, $Ent_{n}\left( f|_{K},\mathcal{U}|_{K}\right)
=Ent_{N}\left( f,\mathcal{U},K\right) $, where $\mathcal{U}|_{K}=\left\{
U\cap K:U\in \mathcal{U}\right\} $ and mapping $f|_{K}:K\rightarrow K$ is
defined by $f|_{K}\left( k\right) =f\left( k\right) $ for each $k\in K$.
\end{theorem}

\begin{proof}
Let $N_{n}\left( \bigvee_{i=0}^{m-1}\left( f|_{K}\right) ^{-i}\left(
\mathcal{U}|_{K}\right) \right) =s$. Then there exist a finite subfamily
\left\{ U_{1},U_{2},\ldots ,U_{s}\right\} $ of $\bigvee_{i=0}^{m-1}$ $\left(
f|_{K}\right) ^{-i}\left( \mathcal{U}|_{K}\right) $ such that $K\subseteq
\bigcup_{l=1}^{s}U_{l}$. Then, for each $l\in \left\{ 1,2,\ldots ,s\right\}
, there exist $\left\{ U_{l}^{0},U_{l}^{1},U_{l}^{2},\ldots
,U_{l}^{m-1}\right\} \in \mathcal{U}$ such that $U_{l}=\bigcap_{i=0}^{m-1
\left( f|_{K}\right) ^{-i}\left( U_{l}^{i}\cap K\right) $. Clearly, $\left\{
U_{l}^{^{\prime }}:U_{l}^{^{\prime }}=\bigcap_{i=0}^{m-1}f^{-i}\left(
U_{l}^{i}\right) \right\} $ is a subfamily of $\bigvee_{i=0}^{m-1}f^{-i
\left( \mathcal{U}\right) $ with $U_{l}\subset U_{l}^{^{\prime }}$.
Moreover,
\begin{equation*}
K=\left( \bigcup_{l=1}^{s}U_{l}\right) \cap K=\bigcup_{l=1}^{s}\left(
U_{l}\cap K\right) \subseteq \bigcup_{l=1}^{s}\left( U_{l}^{^{\prime }}\cap
K\right) \subseteq \bigcup_{l=1}^{s}U_{l}^{^{\prime }}.
\end{equation*
Thus,
\begin{equation*}
N_{K}\left( \bigvee_{i=0}^{m-1}f^{-i}\left( \mathcal{U}\right) \right) \leq
s=N_{n}\left( \bigvee_{i=0}^{m-1}\left( f|_{K}\right) ^{-i}\left( \mathcal{U
|_{K}\right) \right)
\end{equation*
and so $M_{K}\left( \bigvee_{i=0}^{m-1}f^{-i}\left( \mathcal{U}\right)
\right) \leq M_{n}\left( \bigvee_{i=0}^{m-1}\left( f|_{K}\right) ^{-i}\left(
\mathcal{U}|_{K}\right) \right) $. Hence
\begin{eqnarray*}
Ent_{N}\left( f,\mathcal{U},K\right) &=&\underset{m\rightarrow \infty }{\lim
}\frac{1}{m}M_{K}\left( \dbigvee\limits_{i=0}^{m-1}f^{-i}\left( \mathcal{U
\right) \right) \\
&\leq &\lim_{m\rightarrow \infty }\frac{1}{m}M_{n}\left(
\bigvee_{i=0}^{m-1}\left( f|_{K}\right) ^{-i}\left( \mathcal{U}|_{K}\right)
\right) \\
&=&Ent_{n}\left( f|_{K},\mathcal{U}|_{K}\right) .
\end{eqnarray*}

To establish the reverse inequality, consider $N_{K}\left(
\bigvee_{i=0}^{m-1}f^{-i}\left( \mathcal{U}\right) \right) =q$. Then there
exists a finite subfamily $\left\{ V_{1},V_{2},\ldots ,V_{q}\right\} $ of
\bigvee_{i=0}^{m-1}f^{-i}\left( \mathcal{U}\right) $ such that $K\subseteq
\bigcup_{j=1}^{q}V_{j}$. Then for each $j\in \left\{ 1,2,\ldots ,q\right\}
, there exist $\left\{ V_{j}^{0},V_{j}^{1},V_{j}^{2},\ldots
,V_{j}^{m-1}\right\} $ $\in \mathcal{U}$ such that $V_{j}
\bigcap_{i=0}^{m-1}f^{-i}\left( V_{j}^{i}\right) $. Consider the subfamily
\begin{equation*}
\left\{ V_{j}^{^{\prime }}:V_{j}^{^{\prime }}=\bigcap_{i=0}^{m-1}\left(
f|_{K}\right) ^{-i}\left( V_{j}^{i}\cap K\right) \right\}
\end{equation*
of $\bigvee_{i=0}^{m-1}\left( f|_{K}\right) ^{-i}\left( \mathcal{U
|_{K}\right) $. Since $K\subseteq \bigcup_{j=1}^{q}V_{j}$, then
\bigcup_{j=1}^{q}\left( V_{j}\cap K\right) =K$. Again, $f\left( K\right)
\subseteq K$ ensure that $K\subseteq f^{-i}\left( K\right) $ for each
i=0,1,2,\ldots $, and
\begin{equation*}
V_{j}\cap K=\bigcap_{i=0}^{m-1}\left( f^{-i}\left( V_{j}^{i}\right) \cap
K\right) \subseteq \bigcap_{i=0}^{m-1}\left( f^{-i}\left( V_{j}^{i}\right)
\cap f^{-i}\left( K\right) \right) =\bigcap_{i=0}^{m-1}f^{-i}\left(
V_{j}^{i}\cap K\right) .
\end{equation*
Thus,
\begin{equation*}
V_{j}\cap K=\left( \bigcap_{i=0}^{m-1}f^{-i}\left( V_{j}^{i}\cap K\right)
\right) \cap K=\bigcap_{i=0}^{m-1}\left( f|_{K}\right) ^{-i}\left(
V_{j}^{i}\cap K\right) =V_{j}^{^{\prime }}
\end{equation*
and so
\begin{equation*}
K=\bigcup_{j=1}^{q}\left( V_{j}\cap K\right)
=\bigcup_{j=1}^{q}V_{j}^{^{\prime }}.
\end{equation*
Hence, $N_{n}\left( \bigvee_{i=0}^{m-1}\left( f|_{K}\right) ^{-i}\left(
\mathcal{U}|_{K}\right) \right) \leq q=N_{K}\left(
\bigvee_{i=0}^{m-1}f^{-i}\left( \mathcal{U}\right) \right) $ and so,
\begin{equation*}
M_{n}\left( \bigvee_{i=0}^{m-1}\left( f|_{K}\right) ^{-i}\left( \mathcal{U
|_{K}\right) \right) \leq M_{K}\left( \bigvee_{i=0}^{m-1}f^{-i}\left(
\mathcal{U}\right) \right) .
\end{equation*
Therefore,
\begin{eqnarray*}
Ent_{n}\left( f|_{K},\mathcal{U}|_{K}\right) &=&\lim_{m\rightarrow \infty
\frac{1}{m}M_{n}\left( \bigvee_{i=0}^{m-1}\left( f|_{K}\right) ^{-i}\left(
\mathcal{U}|_{K}\right) \right) \\
&\leq &\lim_{m\rightarrow \infty }\frac{1}{m}M_{K}\left(
\bigvee_{i=0}^{m-1}f^{-i}\left( \mathcal{U}\right) \right) \\
&=&Ent_{N}\left( f,\mathcal{U},K\right) .
\end{eqnarray*
Thus, $Ent_{n}\left( f|_{K},\mathcal{U}|_{K}\right) =Ent_{N}\left( f
\mathcal{U},K\right) .$
\end{proof}

The following theorem indicates that the concept of topological nearly
entropy $Ent_{N}\left( f\right) $ is coincides with $Ent_{n}\left( f\right) $
when $X$ itself is nearly compact.

\begin{theorem}
Let $\left( X,f\right) $ be a nearly compact $R$-dynamical system. Then
Ent_{n}\left( f\right) =Ent_{N}\left( f\right) $.
\end{theorem}

\begin{proof}
It is clear that $X\in H\left( X,f\right) $ and by Definition \ref{def_N2}
\begin{equation*}
Ent_{N}\left( f\right) =\sup_{K}\left\{ Ent_{N}\left( f,K\right) :K\in
H\left( X,f\right) \right\} =Ent_{N}\left( f,X\right) .
\end{equation*
Again for any regular open cover $\mathcal{U}$ of $X$, Theorem \ref{n=N}
ensures that $Ent_{n}\left( f,\mathcal{U}\right) =Ent_{N}\left( f,\mathcal{U
,X\right) $ and s
\begin{eqnarray*}
Ent_{N}\left( f,X\right) &=&\sup_{\mathcal{U}}\left\{ Ent_{N}\left( f
\mathcal{U},X\right) :\mathcal{U}~\text{is~a~regular~open~cover~of}~X\right\}
\\
&=&\sup_{\mathcal{U}}\left\{ Ent_{n}\left( f,\mathcal{U}\right) :\mathcal{U}
\text{is~a~regular~open~cover~of}~X\right\} \\
&=&Ent_{n}\left( f\right) .
\end{eqnarray*
Thus, $Ent_{N}\left( f\right) =Ent_{N}\left( f,X\right) =Ent_{n}\left(
f\right) $.
\end{proof}

Next , we study some properties of of topological nearly entropy on nearly
compact and Hausdorff space by introducing $R$-space. So, the result
demonstrate that topological nearly entropy of a map $f$ is coincide with
topological nearly entropy of it restriction $f|_{K}$ whenever $K\in H\left(
X,f\right) $.

Recall that in general, the union of two regular open sets is not regular
open. For example, let $\left( 1,2\right) $ and $\left( 2,3\right) $ be two
regular open sets in $\mathbb{R}$. Hence, the union of both intervals are
not regular open since $\limfunc{int}\left( \limfunc{cl}\left( \left(
1,2\right) \cup \left( 2,3\right) \right) \right) =\left( 1,3\right) $ but
\left( 1,3\right) \neq \left( 1,2\right) \cup \left( 2,3\right) $. As an
upshot, we introduce $R$-space as follows.

\begin{definition}
A topological space $X$ is said to be $R$-space if the union of each regular
open subset of $X$ is also regular open.
\end{definition}

Now, by using $R$-space we study some basic properties of nearly compact in
Hausdorff space.

\begin{lemma}
\label{lem haus} Let $X$ be a Hausdorff and $R$-space. If $A$ is a nearly
compact relative to $X$ and $p\notin A$, then there exist regular open sets
G,H$ of $X$ such that $p\in G,A\subset H$ and $G\cap H=\emptyset .$
\end{lemma}

\begin{proof}
Let $a\in A$. Since $p\notin A$, $p\neq a$. By hypothesis, $X$ is Hausdorff,
hence there exist open sets $G_{a},H_{a}$ of $X$ such that $p\in G_{a},a\in
H_{a}$ and $G_{a}\cap H_{a}=\emptyset .$ Since $A\subset \bigcup \left\{
H_{a}:a\in A\right\} $, then $\left\{ H_{a}:a\in A\right\} $ forms a cover
of $A$ by open subsets of $X$. As $A$ is nearly compact relative to $X$, so
there exist a finite subfamily $H_{a_{1}},\ldots ,H_{a_{m}}$ of $\left\{
H_{a}\right\} $ such that $A\subset \bigcup_{i=1}^{m}\limfunc{int}\left(
\limfunc{cl}\left( H_{a_{i}}\right) \right) $.

Let $H=\limfunc{int}\left( \limfunc{cl}\left( H_{a_{1}}\right) \right) \cup
\cdots \cup \limfunc{int}\left( \limfunc{cl}\left( H_{a_{m}}\right) \right) $
and $G=\limfunc{int}\left( \limfunc{cl}\left( G_{a_{1}}\right) \right) \cap
\cdots \cap \limfunc{int}\left( \limfunc{cl}\left( G_{a_{m}}\right) \right)
. Since $X$ is $R$-space, then $H$ and $G$ are regular open. Furthermore,
A\subset H$ and $p\in G$ since $p$ belongs to each $G_{a_{i}}$ individually.
Lastly we claim that $G\cap H=\emptyset $. Note first that, $\limfunc{int
\left( \limfunc{cl}\left( G_{a_{i}}\right) \right) $ $\cap $ $\limfunc{int
\left( \limfunc{cl}\left( H_{a_{i}}\right) \right) =\emptyset $ implies
G\cap \limfunc{int}\left( \limfunc{cl}\left( H_{a_{i}}\right) \right)
=\emptyset $. Thus, by the distributive law
\begin{eqnarray*}
G\cap H &=&G\cap \left( \limfunc{int}\left( \limfunc{cl}\left(
H_{a_{1}}\right) \right) \cup \cdots \cup \limfunc{int}\left( \limfunc{cl
\left( H_{a_{m}}\right) \right) \right) \\
&=&\left( G\cap \limfunc{int}\left( \limfunc{cl}\left( H_{a_{1}}\right)
\right) \right) \cup \cdots \cup \left( G\cap \limfunc{int}\left( \limfunc{c
}\left( H_{a_{m}}\right) \right) \right) \\
&=&\emptyset \cup \cdots \cup \emptyset \\
&=&\emptyset .
\end{eqnarray*
This completes the proof.
\end{proof}

\begin{lemma}
\label{lem haus2} Let $X$ be a Hausdorff and $R$-space. If $A$ is a nearly
compact relative to $X$ and $p\notin A$, then there is a regular open set $G$
such that $p\in G\subset A^{c}$.
\end{lemma}

\begin{proof}
By Lemma \ref{lem haus}, there exist regular open sets $G$ and $H$ in $X$
such that $p\in G,A\subset H$ and $G\cap H=\emptyset $. Hence $G\cap
A=\emptyset $, and thus $p\in G\subset A^{c}.$
\end{proof}

\begin{lemma}
\label{lem haus3} Let $X$ be a Hausdorff and $R$-space. If $A$ is a nearly
compact relative to $X$, then $A$ is a regular closed subset of $X$.
\end{lemma}

\begin{proof}
We prove, equivalently, that $A^{c}$ is a regular open subset of $X$. Let
p\in A^{c}$, i.e., $p\notin A$. Then by Lemma \ref{lem haus2} there exists a
regular open set $G_{p}$ of $X$ such that $p\in G_{p}\subset A^{c}$. Hence
A^{c}=\bigcup_{p\in A^{c}}G_{p}$. Thus $A^{c}$ is regular open as it is the
union of regular open sets in $R$-space $X$. Therefore $A$ is a regular
closed subset of $X$.
\end{proof}

The next theorem shows that the conditions on which the topological nearly
entropy of $f$ and it restriction $f|_{K}$ coincides.

\begin{theorem}
Let $\left( X,f\right) $ be a topological $R$-dynamical system. Assume that
X$ is Hausdorff and $R$-space. If $K\in H\left( X,f\right) $ then

$\left( a\right) $ $Ent_{N}\left( f,K\right) =Ent_{n}\left( f|_{K},K\right) $

$\left( b\right) $ $Ent_{N}\left( f\right) =Ent_{n}\left( f|_{K}\right) ,$

where $f|_{K}$ is defined as in the Theorem \ref{n=N}.
\end{theorem}

\begin{proof}
$\left( a\right) $ Let $\mathcal{U}_{K}$ be any regular open cover of $K$
(by regular open subsets of $K$). Since $X$ is Hausdorff and $K$ is nearly
compact relative to $X$, then by Lemma \ref{lem haus3}, $K$ is a regular
closed subset of $X$. For every $A\in \mathcal{U}_{K}$, there exist a
regular open subset $U_{A}$ of $X$ satisfying $A=U_{A}\cap K$. Denote
\mathcal{U}^{\prime }=\left\{ U_{A}:A\in \mathcal{U}_{K}\right\} $. Clearly,
$\mathcal{U}=\mathcal{U}^{\prime }\cup \left\{ X\setminus K\right\} $ is a
regular open cover of $X$ satisfying $\mathcal{U}|_{K}=\mathcal{U}_{K}\cup
\left\{ \emptyset \right\} $. Hence, every regular open cover $\mathcal{U
_{K}$ of $K$ is a restriction $\mathcal{U}|_{K}$ of some special regular
open cover $\mathcal{U}$ of $X$ that includes regular open subset
X\setminus K$ of $X$. From Definition \ref{def topo N}, we have
\begin{equation}
Ent_{n}\left( f|_{K},K\right) =\sup_{\mathcal{U}_{K}}\left\{ Ent_{n}\left(
f|_{K},\mathcal{U}_{K},K\right) \right\} =\sup_{\mathcal{U}}\left\{
Ent_{n}\left( f|_{K},\mathcal{U}|_{K},K\right) \right\} .  \label{2}
\end{equation
By Definition \ref{def_N1}, we have
\begin{equation*}
Ent_{N}\left( f,K\right) =\sup_{\beta }\left\{ Ent_{N}\left( f,\beta
,K\right) :\beta \text{ runs over all regular open covers of }X\right\} .
\end{equation*
It follows from Theorem \ref{n=N} that, $Ent_{N}\left( f,\mathcal{U
,K\right) =Ent_{n}\left( f|_{K},\mathcal{U}|_{K},K\right) $, thus
Ent_{n}\left( f|_{K},K\right) =\sup_{\mathcal{U}}\left\{ Ent_{n}\left(
f|_{K},\mathcal{U}|_{K},K\right) \right\} =\sup_{\mathcal{U}}$ $\left\{
Ent_{N}\left( f,\mathcal{U},K\right) \right\} $. Recall that the regular
open covers $\mathcal{U}$ are some special regular open covers of $X$.
Hence,
\begin{equation*}
\sup_{\mathcal{U}}\left\{ Ent_{N}\left( f,\mathcal{U},K\right) \right\} \leq
Ent_{N}\left( f,K\right)
\end{equation*
implying
\begin{equation*}
Ent_{n}\left( f|_{K},K\right) \leq Ent_{N}\left( f,K\right) .
\end{equation*}

For reverse inequality, i.e., $Ent_{n}\left( f|K,K\right) \geq Ent_{N}\left(
f,K\right) $, let $\mathcal{U}$ be any regular open cover of $X$. From
Theorem \ref{n=N}, we have $Ent_{N}\left( f,\mathcal{U},K\right)
=Ent_{n}\left( f|_{K},\mathcal{U}|_{K},K\right) $ and thus from Definition
\ref{def_N1}
\begin{equation*}
Ent_{N}\left( f,K\right) =\sup_{\mathcal{U}}\left\{ Ent_{N}\left( f,\mathcal
U},K\right) \right\} =\sup_{\mathcal{U}}\left\{ Ent_{n}\left( f|_{K}
\mathcal{U}|_{K},K\right) \right\} .
\end{equation*
As $\mathcal{U}|_{K}$ are some special regular open covers of $K$, we hav
\begin{equation*}
\sup_{\mathcal{U}}\left\{ Ent_{n}\left( f|_{K},\mathcal{U}|_{K},K\right)
\right\} \leq Ent_{n}\left( f|_{K},K\right) .
\end{equation*
Thus
\begin{eqnarray*}
Ent_{N}\left( f,K\right) &=&\sup_{\mathcal{U}}\left\{ Ent_{N}\left( f
\mathcal{U},K\right) \right\} \\
&=&\sup_{\mathcal{U}}\left\{ Ent_{n}\left( f|_{K},\mathcal{U}|_{K},K\right)
\right\} \\
&\leq &Ent_{n}\left( f|_{K},K\right) .
\end{eqnarray*}

$\left( b\right) $ By Definition \ref{def_N2}
\begin{equation*}
Ent_{N}\left( f\right) =\sup_{K}\left\{ Ent_{N}\left( f,K\right) :K\in
H\left( X,f\right) \right\} .
\end{equation*
Now let $\mathcal{U}$ be any regular open cover of $X$. From Definition \re
{def_N1},
\begin{equation*}
Ent_{N}\left( f,K\right) =\sup_{\mathcal{U}}\left\{ Ent_{N}\left( f,\mathcal
U},K\right) :\mathcal{U}\mathrm{\ }\text{is~a regular~open~cover of
~X\right\} .
\end{equation*
Thus
\begin{eqnarray*}
Ent_{N}\left( f\right) &=&\sup_{K}\left\{ \sup_{\mathcal{U}}\left\{
Ent_{N}\left( f,\mathcal{U},K\right) :\mathcal{U}\mathrm{\ }\text{is~a
regular~open~cover of}~X\right\} :K\in H\left( X,f\right) \right\} \\
&=&\sup_{\mathcal{U}}\left\{ \sup_{K}\left\{ Ent_{N}\left( f,\mathcal{U
,K\right) :K\in H\left( X,f\right) \right\} :\mathcal{U}\mathrm{\ }\text
is~a regular~open~cover of}~X\right\} \\
&=&\sup_{\mathcal{U}}\left\{ Ent_{N}\left( f,\mathcal{U}\right) :\text{
\mathcal{U}\mathrm{\ }\text{is~a regular~open~cover of}~X\right\}
\end{eqnarray*
By Theorem \ref{n=N},
\begin{equation*}
Ent_{N}\left( f,\mathcal{U}\right) =Ent_{n}\left( f|_{K},\mathcal{U
|_{K}\right) .
\end{equation*
Hence
\begin{eqnarray*}
Ent_{N}\left( f\right) &=&\sup_{\mathcal{U}}\left\{ Ent_{n}\left( f|_{K}
\mathcal{U}|_{K}\right) :\text{ }\mathcal{U}\mathrm{\ }\text{is~a
regular~open~cover of}~X\right\} \\
&=&\sup_{\mathcal{U}_{K}}\left\{ Ent_{n}\left( f|_{K},\mathcal{U
_{K},K\right) :\mathcal{U}_{K}\mathrm{\ }\text{is~a regular~open~cover of
~K\right\} \\
&=&Ent_{n}\left( f|_{K}\right) .
\end{eqnarray*
The second equality follows from equation $\left( \ref{2}\right) $. This
completes the proof.
\end{proof}

\section{Topological Nearly Entropy of Product Space}

Let $\left( X,f\right) $ and $\left( Y,h\right) $ be two topological $R
-dynamical systems. For the product space $X\times Y$, define function
f\times h:X\times Y\rightarrow X\times Y$ by $\left( f\times h\right) \left(
x,y\right) =\left( f\left( x\right) ,h\left( y\right) \right) $. This
function $f\times h$ is $R$-map and $\left( X\times Y,f\times h\right) $
forms a topological $R$-dynamical system. If $\mathcal{U}$ and $\mathcal{V}$
are regular open covers of $X$ and $Y$, respectively, then $\mathcal{U
\times \mathcal{V}$ is a regular open cover of $X\times Y$.

As a consequence, we obtain the properties of topological nearly entropy for
product space. But, firstly we show that $R$-map preserve nearly compactness
relative to the space.

\begin{lemma}
\label{fA} Let $f:X\rightarrow Y$ be an $R$-map. If $A$ is nearly compact
relative to $X$, then $f\left( A\right) $ is nearly compact relative to $Y$.
\end{lemma}

\begin{proof}
Let $\left\{ V_{\alpha }:\alpha \in \Delta \right\} $ be a cover of $f\left(
A\right) $ by regular open subsets of $Y$. For each $x\in A$, take $\alpha
_{x}\in \Delta $ such that $f\left( x\right) \in V_{\alpha _{x}}$. Since $f$
is $R$-map, $f^{-1}\left( V_{\alpha _{x}}\right) $ is a regular open subset
of $X$ containing $x$. Now $\left\{ f^{-1}\left( V_{\alpha _{x}}\right)
:x\in A\right\} $ forms a regular open cover of $A$ by regular open subsets
of $X$. Since $A$ is nearly compact relative to $X$, there exists a finite
subset of points $x_{1},\ldots ,x_{n}$ of $A$ such that $A\subseteq
\bigcup_{k=1}^{n}f^{-1}\left( V_{\alpha _{x_{k}}}\right) $. Now,
\begin{equation*}
f\left( A\right) \subseteq f\left( \bigcup_{k=1}^{n}f^{-1}\left( V_{\alpha
_{x_{k}}}\right) \right) =\bigcup_{k=1}^{n}f\left( f^{-1}\left( V_{\alpha
_{x_{k}}}\right) \right) \subseteq \bigcup_{k=1}^{n}V_{\alpha _{x_{k}}}.
\end{equation*
Therefore $f\left( A\right) $ is nearly compact relative to $Y$.
\end{proof}

\begin{lemma}
\label{cover} Let $X$ and $Y$ be two nearly compact $R$-dynamical system.
Suppose that $\mathcal{U}$ and $\mathcal{V}$ are regular open covers of $X$
and $Y$, respectively. Then for a regular open cover $\mathcal{U}\times
\mathcal{V}$ of $X\times Y$, we hav
\begin{equation*}
M_{n}\left( \mathcal{U}\times \mathcal{V}\right) \leq M_{n}\left( \mathcal{U
\right) +M_{n}\left( \mathcal{V}\right) .
\end{equation*}
\end{lemma}

\begin{proof}
Let $\left\{ U_{1},\ldots ,U_{k}\right\} $ and $\left\{ V_{1},\ldots
,V_{p}\right\} $ be subcovers of $\mathcal{U}$ and $\mathcal{V}$ of minimal
cardinality, respectively. Then $\left\{ U_{i}\times V_{j}:i=1,\ldots
,k,j=1,\ldots ,p\right\} $ is a subcover of $\mathcal{U}\times \mathcal{V}$
with cardinality $kp$. It follows tha
\begin{equation*}
N_{n}\left( \mathcal{U}\times \mathcal{V}\right) \leq kp=N_{n}\left(
\mathcal{U}\right) \cdot N_{n}\left( \mathcal{V}\right) .
\end{equation*
Thus
\begin{eqnarray*}
M_{n}\left( \mathcal{U}\times \mathcal{V}\right) &=&\log N_{n}\left(
\mathcal{U}\times \mathcal{V}\right) \\
&\leq &\log \left( N_{n}\left( \mathcal{U}\right) \cdot N_{n}\left( \mathcal
V}\right) \right) \\
&=&\log N_{n}\left( \mathcal{U}\right) +\log N_{n}\left( \mathcal{V}\right)
\\
&=&M_{n}\left( \mathcal{U}\right) +M_{n}\left( \mathcal{V}\right) .
\end{eqnarray*}
\end{proof}

\begin{lemma}
\label{lem 0.1} Let $\left( X,f\right) $ and $\left( Y,h\right) $ be two
topological $R$-dynamical systems. Suppose $T_{x}:X\times Y\rightarrow X$
and $T_{y}:X\times Y\rightarrow Y$ are the $R$-map projections on $X$ and $Y
, respectively. If $K\in H\left( X\times Y,f\times h\right) $, then
T_{x}\left( K\right) \in H\left( X,f\right) $, $T_{y}\left( K\right) \in
H\left( Y,h\right) $ and $K\subseteq T_{x}\left( K\right) \times T_{y}\left(
K\right) $.
\end{lemma}

\begin{proof}
Since $T_{x}$ and $T_{y}$ are $R$-maps and $K\in H\left( X\times Y,f\times
h\right) $, $T_{x}\left( K\right) $ is nearly compact relative to $X$ and
T_{y}\left( K\right) $ is nearly compact relative to $Y$ by Lemma \ref{fA}.
It suffices to show that $T_{x}\left( K\right) $ and $T_{y}\left( K\right) $
are invariant nearly compact relative to $X$ and $Y$ by $f$ and $h$,
respectively. For any $x\in T_{x}\left( K\right) $, there exists a $y\in Y$
which satisfying $\left( x,y\right) \in K$. Since $K$ is an invariant nearly
compact relative to $X\times Y$ by $f\times h$, we have
\begin{equation*}
\left( f\times h\right) \left( x,y\right) =\left( f\left( x\right) ,h\left(
y\right) \right) \in K.
\end{equation*
Thus
\begin{equation*}
T_{x}\left( f\left( x\right) ,h\left( y\right) \right) =f\left( x\right) \in
T_{x}\left( K\right) .
\end{equation*
Hence $f\left( T_{x}\left( K\right) \right) \subseteq T_{x}\left( K\right) $
and therefore $T_{x}\left( K\right) \in H\left( X,f\right) $.

There also exists an $x\in X$ for any $y\in T_{y}\left( K\right) $
satisfying $\left( x,y\right) \in K$. Since $K$ is an invariant nearly
compact relative to $X\times Y$ by $f\times h$, we have
\begin{equation*}
\left( f\times h\right) \left( x,y\right) =\left( f\left( x\right) ,h\left(
y\right) \right) \in K.
\end{equation*
Thus
\begin{equation*}
T_{y}\left( f\left( x\right) ,h\left( y\right) \right) =h\left( y\right) \in
T_{y}\left( K\right) .
\end{equation*
Hence $h\left( T_{y}\left( K\right) \right) \subseteq T_{y}\left( K\right) $
and therefore $T_{y}\left( K\right) \in H\left( Y,h\right) .$

Finally, if $\left( x,y\right) \in K$, then by the definitions of
T_{x}\left( K\right) $ and $T_{y}\left( K\right) $, we will have $x\in
T_{x}\left( K\right) $ and $y\in T_{y}\left( K\right) $ and thus $\left(
x,y\right) \in $ $T_{x}\left( K\right) \times T_{y}\left( K\right) $.
\end{proof}

\begin{lemma}
\label{lem 1}\cite{singal} The product of two nearly compact spaces is
nearly compact.
\end{lemma}

\begin{lemma}
\label{lem 0.3} Let $\left( X,f\right) $ and $\left( Y,h\right) $ be two
nearly compact $R$-dynamical systems. If $\mathcal{W}$ is a regular open
cover of $X\times Y$, then there exist regular open cover $\mathcal{U}$ of
X $ and regular open cover $\mathcal{V}$ of $Y$ satisfying $\mathcal{W}\prec
\mathcal{U}\times \mathcal{V}$.
\end{lemma}

\begin{proof}
Since $X$ and $Y$ are nearly compact, by Lemma \ref{lem 1} the product space
of $X\times Y$ is also nearly compact and for each $x\in X$, $\left\{
x\right\} \times Y$ is a nearly compact subset of $X\times Y$. Since
\mathcal{W}$ is a regular open cover of $X\times Y$, then there exist
A_{y}\in \mathcal{W}$ for any $y\in Y$ which satisfying $\left( x,y\right)
\in A_{y}$. Let $U\left( x,y\right) $ be a regular open neighborhood of $x$
in $X$ and $V\left( x,y\right) $ be a regular open neigborhood of $y$ in $Y$
wit
\begin{equation*}
U\left( x,y\right) \times V\left( x,y\right) \subseteq A_{y}.
\end{equation*
Since $Y$ is nearly compact, the regular open cover $\left\{ V\left(
x,y\right) :y\in Y\right\} $ of $Y$ has a finite subcover, say $\left\{
V\left( x,y_{1}\right) ,\ldots ,V\left( x,y_{m\left( x\right) }\right)
\right\} $. Hence, for each $U\left( x,y_{j}\right) \times V\left(
x,y_{j}\right) ,j\in \left\{ 1,2,\ldots ,m\left( x\right) \right\} $, there
exist an $A_{j}\in \mathcal{W}$ such that $U\left( x,y_{j}\right) \times
V\left( x,y_{j}\right) \subseteq A_{j}$. Denote $U\left( x\right)
=\dbigcap\limits_{j=1}^{m\left( x\right) }U\left( x,y_{j}\right) $, the
\begin{equation*}
U\left( x\right) \times V\left( x,y_{j}\right) \subseteq A_{j}.
\end{equation*
On the other hand, as $X$ is nearly compact, the regular open cover $\left\{
U\left( x\right) :x\in X\right\} $ of $X$ has a finite subcover, say
\left\{ U\left( x_{1}\right) ,\ldots ,U\left( x_{n}\right) \right\} $.
Observe that each $U\left( x_{i}\right) \times V\left( x_{i},y_{j}\right)
,i\in \left\{ 1,2,\ldots ,n\right\} ,j\in \left\{ 1,2,\ldots ,m\left(
x_{i}\right) \right\} $ is contained in an element of $\mathcal{W}$. For any
$y\in Y$, let $V\left( y\right) $ be the intersection of those elements of
the collection $\left\{ V\left( x_{i},y_{j}\right) :i=1,\ldots ,n;j=1,\ldots
,m\left( x_{i}\right) \right\} $ that contain $y$. Since this collection has
finitely many elements and each of them is regular open subset of $Y$, then
V\left( y\right) $ is also regular open subset of $Y$. Notice that $\left\{
V\left( y\right) :y\in Y\right\} $ is a regular open cover of $Y$ and hence
it has a finite subcover as $Y$ is nearly compact. Denote this subcover by
\left\{ V\left( y_{1}\right) ,\ldots ,V\left( y_{m}\right) \right\} $. Now,
\left\{ U\left( x_{i}\right) \times V\left( y_{j}\right) :i=1,\ldots
,n;j=1,\ldots ,m\right\} $ is a regular open cover of $X\times Y$ and each
U\left( x_{i}\right) \times V\left( y_{j}\right) $ is contained in an
element of $\mathcal{W}$. Let $\mathcal{U}=\left\{ U\left( x_{i}\right)
:i=1,\ldots ,n\right\} $ and $\mathcal{V}=\left\{ V\left( y_{j}\right)
:j=1,\ldots ,m\right\} $. Then $\mathcal{U}\times \mathcal{V}$ is a regular
open cover of $X\times Y$ and $\mathcal{W}\prec \mathcal{U}\times \mathcal{V}
$.
\end{proof}

\begin{theorem}
\label{Th 4.2}Let $\left( X,f\right) $ and $\left( Y,h\right) $ be two
nearly compact $R$-dynamical systems. Then
\begin{equation*}
Ent_{n}\left( f\times h\right) \leq Ent_{n}\left( f\right) +Ent_{n}\left(
h\right) ,
\end{equation*
where $f\times h:X\times Y\rightarrow X\times Y$ is $R$-map defined by
\left( f\times h\right) \left( x,y\right) =\left( f\left( x\right) ,h\left(
y\right) \right) $.
\end{theorem}

\begin{proof}
Let $\mathcal{C}$ be an arbitrary regular open cover of $X\times Y$. By
Lemma \ref{lem 0.3}, there exist regular open cover $\mathcal{U}$ of $X$ and
regular open cover $\mathcal{V}$ of $Y$ such that $\mathcal{U
=\{U_{i}:i=1,\ldots ,n\}$ where $U_{i}$ is a regular open subset of $X$ and
\mathcal{V}=\{V_{j}:j=1,\ldots ,m\}$ where $V_{j}$ is a regular open subset
of $Y$ satisfying\ $\mathcal{C}\prec \mathcal{U}\times \mathcal{V}$. By
Theorem \ref{Th2.1} and Lemma \ref{cover}
\begin{eqnarray*}
Ent_{n}\left( f\times h,\mathcal{C}\right) &\leq &Ent_{n}\left( f\times h
\mathcal{U}\times \mathcal{V}\right) \\
&\leq &Ent_{n}\left( f,\mathcal{U}\right) +Ent_{n}\left( h,\mathcal{V
\right) .
\end{eqnarray*
So,
\begin{eqnarray*}
Ent_{n}\left( f\times h\right) &=&\sup_{\mathcal{C}}\{Ent_{n}\left\{ f\times
h,\mathcal{C}\right\} \\
&\leq &\sup_{\mathcal{C}}\left\{ Ent_{n}\left( f,\mathcal{U}\right)
+Ent_{n}\left( h,\mathcal{V}\right) \right\} \\
&\leq &\sup_{\mathcal{U}}\left\{ Ent_{n}\left( f,\mathcal{U}\right) \right\}
+\sup_{\mathcal{V}}\left\{ Ent_{n}\left( h,\mathcal{V}\right) \right\} \\
&=&Ent_{n}\left( f\right) +Ent_{n}\left( h\right) .
\end{eqnarray*}
\end{proof}

\begin{theorem}
Let $\left( X,f\right) $ and $\left( Y,h\right) $ be two topological $R
-dynamical systems with $X\times Y$ are $R$-space. If $X$ and $Y$ are
Hausdorff, then $Ent_{N}\left( f\times h\right) \leq Ent_{N}\left( f\right)
+Ent_{N}\left( h\right) $, where $f\times h:X\times Y\rightarrow X\times Y$
is $R$-map defined by $\left( f\times h\right) \left( x,y\right) =\left(
f\left( x\right) ,h\left( y\right) \right) $.
\end{theorem}

\begin{proof}
If $H\left( X\times Y,f\times h\right) =\emptyset $, then the result follows
since $Ent_{N}\left( f\times h\right) =0$ by Definition \ref{def_N2}.

If $H\left( X\times Y,f\times h\right) \neq \emptyset $, let $K\in H\left(
X\times Y,f\times h\right) $ and the projections $T_{x}:X\times Y\rightarrow
X$, $T_{y}:X\times Y\rightarrow Y$. For any regular open cover $\mathcal{W}$
of $X\times Y$, $T_{x}\left( K\right) \in H\left( X,f\right) ,T_{y}\left(
K\right) \in H\left( X,f\right) $ and $K\subseteq T_{x}\left( K\right)
\times T_{y}\left( K\right) $ by Lemma \ref{lem 0.1}. Since $X$ and $Y$ are
Hausdorff and $R$-spaces, then by Lemmas \ref{fA} and \ref{lem haus3}
T_{x}\left( K\right) $ and $T_{y}\left( K\right) $ are regular closed subset
of $X$ and $Y$, respectively.

Denote $T_{x}\left( K\right) $ and $T_{y}\left( K\right) $ as $K_{x}$ and
K_{y}$, respectively. Thus $K\subseteq K_{x}\times K_{y}$\ and $K_{x}\times
K_{y}\in H\left( X\times Y,f\times h\right) $ by Lemma \ref{lem 1}.\ By
Theorem \ref{Th 2.2},
\begin{equation*}
Ent_{N}\left( f\times h,\mathcal{W},K\right) \leq Ent_{N}\left( f\times h
\mathcal{W},K_{x}\times K_{y}\right) .
\end{equation*
From Theorem \ref{n=N}, we hav
\begin{equation}
Ent_{N}\left( f\times h,\mathcal{W},K_{x}\times K_{y}\right) =Ent_{n}\left(
\left( f\times h\right) |_{K_{x}\times K_{y}},\mathcal{W}|_{K_{x}\times
K_{y}}\right)  \label{(1)}
\end{equation
where $\mathcal{W}|_{K_{x}\times K_{y}}$ is a regular open cover of
K_{x}\times K_{y}$. By Lemma \ref{lem 0.3} , there exist a regular open
cover of $\mathcal{U^{\prime }}$ of $K_{x}$ and a regular open cover
\mathcal{V^{\prime }}$ of $K_{y}$ such that $\mathcal{U^{\prime }}=\left\{
U_{i}:i=1,\ldots ,n\right\} $ where $U_{i}$ is a regular open subset of
K_{x}$ and $\mathcal{V^{\prime }}=\left\{ V_{j}:j=1,\ldots ,n\right\} $
where $V_{j}$ is a regular open subset of $K_{y}$ satisfying $\mathcal{W
|_{K_{x}\times K_{y}}\prec \mathcal{U^{\prime }}\times \mathcal{V^{\prime }}
. For every $U_{i}$, there exist a regular open subset $A_{i}$ of $X$ such
that $U_{i}=A_{i}\cap K_{x}$. Denote $\mathcal{U}=\left\{ A_{1},\ldots
,A_{n},X\setminus K_{x}\right\} $. Hence $\mathcal{U}$ is regular open cover
of $X$ and $\mathcal{U}|_{K_{x}}=\mathcal{U^{\prime }}\cup \left\{ \emptyset
\right\} $. Similarly, there exist a regular open cover $\mathcal{V}$ of $Y$
such that $\mathcal{V}|_{K_{y}}=\mathcal{V^{\prime }}\cup \left\{ \emptyset
\right\} $. Hence, from $\mathcal{W}|_{K_{x}\times K_{y}}\prec \mathcal
U^{\prime }}\times \mathcal{V^{\prime }}$, we hav
\begin{equation*}
\mathcal{W}|_{K_{x}\times K_{y}}\prec \left( \mathcal{U^{\prime }}\cup
\left\{ \emptyset \right\} \right) \times \left( \mathcal{V^{\prime }}\cup
\left\{ \emptyset \right\} \right) .
\end{equation*
Therefore, we have $\mathcal{W}|_{K_{x}\times K_{y}}\prec \mathcal{U
|_{K_{x}}\times \mathcal{V}|_{K_{y}}$. Moreover, we also hav
\begin{equation}
(\mathcal{U}\times \mathcal{V})|_{K_{x}\times K_{y}}=\mathcal{U
|_{K_{x}}\times \mathcal{V}|_{K_{y}}.  \label{eq1}
\end{equation
By Theorem \ref{thm_n2.1}, equation (\ref{eq1}) and Theorem \ref{Th 4.2}, we
hav
\begin{eqnarray*}
Ent_{n}\left( \left( f\times h\right) |_{K_{x}\times K_{y}},\mathcal{W
|_{K_{x}\times K_{y}}\right) &\leq &Ent_{n}\left( \left( f\times h\right)
|_{K_{x}\times K_{y}},\mathcal{U}|_{K_{x}}\times \mathcal{V}|_{K_{y}}\right)
\\
&=&Ent_{n}\left( \left( f\times h\right) |_{K_{x}\times K_{y}},(\mathcal{U
\times \mathcal{V})|_{K_{x}\times K_{y}}\right) \\
&=&Ent_{n}\left( f|_{K_{x}},\mathcal{U}|_{K_{x}}\right) +Ent_{n}\left(
h|_{K_{y}},\mathcal{V}|_{K_{y}}\right) .
\end{eqnarray*
By Theorem \ref{n=N}
\begin{equation*}
Ent_{N}\left( f\times h,\mathcal{U}\times \mathcal{V},K_{x}\times
K_{y}\right) =Ent_{n}\left( \left( f\times h\right) |_{K_{x}\times
K_{y}},\left( \mathcal{U}\times \mathcal{V}\right) |_{K_{x}\times
K_{y}}\right) .
\end{equation*
Hence, $Ent_{N}\left( f,\mathcal{U},K_{x}\right) =Ent_{n}\left( f|_{K_{x}}
\mathcal{U}|_{K_{x}}\right) $ and $Ent_{N}\left( h,\mathcal{V},K_{y}\right)
=Ent_{n}\left( h|_{K_{y}},\mathcal{V}|_{K_{y}}\right) $. Thus from equation
\left( \ref{(1)}\right) $
\begin{eqnarray*}
Ent_{N}\left( f\times h,\mathcal{U}\times \mathcal{V},K_{x}\times
K_{y}\right) &\leq &Ent_{n}\left( f|_{K_{x}},\mathcal{U}|_{K_{x}}\right)
+Ent_{n}\left( h|_{K_{y}},\mathcal{V}|_{K_{y}}\right) \\
&=&Ent_{N}\left( f,\mathcal{U},K_{x}\right) +Ent_{N}\left( h,\mathcal{V
,K_{y}\right) .
\end{eqnarray*
By Theorem \ref{Th 2.2} and equation $\left( \ref{(1)}\right) $
\begin{eqnarray*}
Ent_{N}\left( f\times h,\mathcal{W},K\right) &\leq &Ent_{N}\left( f\times h
\mathcal{W},K_{x}\times K_{y}\right) \\
&=&Ent_{n}\left( \left( f\times h\right) |_{K_{x}\times K_{y}},\mathcal{W
|_{K_{x}\times K_{y}},K_{x}\times K_{y}\right) \\
&\leq &Ent_{n}\left( f|_{K_{x}},\mathcal{U}|_{K_{x}},K_{x}\right)
+Ent_{n}\left( h|_{K_{y}},\mathcal{V}|_{K_{y}},K_{y}\right) \\
&=&Ent_{N}\left( f,\mathcal{U},K_{x}\right) +Ent_{N}\left( h,\mathcal{V
,K_{y}\right) \\
&\leq &Ent_{N}\left( f\right) +Ent_{N}\left( h\right) .
\end{eqnarray*
Finally by Definitions \ref{def_N1} and \ref{def_N2}, we hav
\begin{eqnarray*}
Ent_{N}\left( f\times h\right) &=&\sup_{K}\left\{ \sup_{\mathcal{W}}\left\{
Ent_{N}\left( f\times h,\mathcal{W},K\right) \right\} \right\} \\
&\leq &Ent_{N}\left( f\right) +Ent_{N}\left( h\right) .
\end{eqnarray*}
\end{proof}

In the next example, we shows $Ent_{N}\left( f\right) =0$ for $f\left(
x\right) =kx$ on
\mathbb{R}
$ with usual topology.

\begin{example}
Let a map $f:\mathbb{R}\rightarrow \mathbb{R}$ define by $f\left( x\right)
=kx$, where $k\in \left\{ 2,3,4,\ldots \right\} $. The only invariant
compact subset of
\mathbb{R}
$ and hence nearly compact subset of
\mathbb{R}
$ is $\left\{ 0\right\} $, so $H\left( \mathbb{R},f\right) =\left\{ \left\{
0\right\} \right\} $. Denote $K=\left\{ 0\right\} $, thus for any regular
open cover $\mathcal{U}$ of $\mathbb{R}$, any subcover $\mathcal{V}$ of
\mathcal{U}$ such that $K\subseteq \dbigcup\limits_{V\in \mathcal{V}}V$ with
the smallest cardinality contains a single element of $\mathcal{U}$, i.e.,
\left\{ 0\right\} $. Hence, $N_{K}\left( \mathcal{U}\right) =1$, so that
N_{K}\left( \bigvee_{i=0}^{n-1}f^{-i}\left( \mathcal{U}\right) \right) =1$
and $M_{K}\left( \bigvee_{i=0}^{n-1}f^{-i}\left( \mathcal{U}\right) \right)
=0$ which implie
\begin{equation*}
Ent_{N}\left( f,\mathcal{U},K\right) =\lim_{n\rightarrow \infty }\frac{1}{n
M_{K}\left( \bigvee_{i=0}^{n-1}f^{-i}\left( \mathcal{U}\right) \right) =0.
\end{equation*
Therefore, from Definition \ref{def_N2}, $Ent_{N}\left( f\right) =0$. If we
replaced $\mathbb{R}$ by $\mathbb{R}^{+}$, then $H\left( \mathbb{R
^{+},f\right) =\emptyset $. Again from Definition \ref{def_N2},
Ent_{N}\left( f\right) =0$. Recall that the role of topological entropy is
to describe the complexity of a topological dynamical system and positivity
of entropy implies the complexity of dynamical behaviour of the system.
Thus, we could say that topological nearly entropy is describe the
complexity of topological $R$-dynamical system and it positive value shows
that more complicated $R$-dynamical properties of the system. Hence, we
conclude that $Ent_{N}\left( f\right) =0$ is a characterization for $R
-dynamical property of the map $f\left( x\right) =kx$.
\end{example}

\section{Conclusion}

We present a notion of topological nearly entropy on nearly compact spaces
and investigate some its properties which is an extension from our previous
paper \cite{gulam}. Both of the notions are coincide in the special case of
nearly compact spaces. Yet, they retain various fundamental properties of
Adler et al. \cite{adler}. A new space, namely, $R$-space was introduced for
nearly compact on Hausdorff space. Consequently, the properties of
topological nearly entropy for product spaces are obtained.

redpajama_set_name: RedPajamaArXiv


text:
NSAC's New Website
Advocacy Champions
We Will Wait and See What the Pope's New Tribunal Will Do
National Survivor Advocates Coalition (NSAC) Statement Regarding Pope Francis' Acceptance of Papal Commission's Recommendations Regarding Bishops and Sexual Abuse
Contact: Kristine Ward, Chair, National Survivor Advocates Coalition (NSAC) 937-272-0308, KristineWard@hotmail.com
The proof is in the pudding.
We will wait and see.
Trusting in this action will require hope. We will hold out hope that The Congregation for the Doctrine of Faith will be capable of a 180 degree reversal of the thinking that permitted, indeed appeared to encourage Bishop Robert Finn, recently resigned from the Diocese of Kansas City-St. Joseph from presiding at ordinations.
If this truly is a back to the drawing board moment, we will watch to see what the new picture looks like.
Sadly, though, there is no indication in today's news that there will be any action taken retroactively against any Bishop regardless of the preponderance of evidence that has surfaced in the crisis. That's a pity. Through depositions and other legal disclosures it has certainly become evident that Cardinals, Archbishops and Bishops have protected abusers and caused children to suffer.
Ultimately, it is Pope Francis' responsibility to remove a Bishop and we hope he does so when a Bishop's actions have protected perpetrators and caused suffering to children and the adults they become.
The National Survivor Advocates Coalition (NSAC), based in Dayton, Ohio in the United States is a confederacy of in the pew Catholics and men and women of good will engaged in educating society regarding sexual abuse, promoting legislation to aid survivors, and working for justice for survivors in the church and in the civil society.
This entry was posted in Editorials, Kristine Ward, Press Release and tagged Pope Francis, Rome, sexual abuse, survivors, Vatican on June 10, 2015 by nsac1.
Tin Ears, Hard Hearts, and Hubris
The National Survivor Advocates Coalition (NSAC) sent letters to 55 Bishops—Archbishops, Bishops, Auxiliary Bishops and Retired Bishops – in Missouri, Kansas, Indiana, Ohio, and Illinois in search of one who would preside in the place of convicted and now resigned Bishop Robert Finn at the Diocese of Kansas City – St. Joseph scheduled May 23rd ordination of candidates for the priesthood.
We offered to pay the transportation expenses for any one of them willing to stand and prevent this travesty from happening.
We are publishing the full text of our letter at this end of this editorial.
Because we have watched the behavior of Bishops in the 13 years since the eruption of the Boston incarnation of the scandal and learned even more from their depositions and dealings with survivors and their families, we know better than to be hopeful that another Bishop will preside in Finn's stead on Saturday.
For that reason, we included in our letter a second request of the 55 Bishops: that unwilling or unable to preside at the Diocese of Kansas City-St. Joseph's ordinations, the Bishops at least contact the temporary administrator of the Diocese of Kansas City-St. Joseph, Bishop Joseph Naumann and request that he change the time so that he could perform the ordinations or incorporate the Diocese of Kansas City-St. Joseph seminarians into the ordination ceremony in his diocese (Diocese of Kansas City, Kansas).
Finn presiding at these ordinations is egregious.
For that, in addition to Bishop Naumann, Cardinal Marc Ouellet, Prefect of the Congregation for Bishops, Archbishop Vigano, the Vatican's Apostolic Nuncio (ambassador) to the United States, both of whom approved of this arrangement at the time of Finn's "voluntary" resignation, April 21, 2015, are responsible.
But ultimately, Pope Francis bears responsibility.
For all the lovely words that have been said about healing, protecting, getting it and moving on in the sexual abuse scandal, – the largest crisis in the Roman Catholic Church in 500 years – this action, allowed to stand, shows the Church's true colors.
We hope those who plan to be cheering in the streets of the United States' cities where Pope Francis will travel in September will remember this Saturday in May.
Beyond that, we do have to ask: why are Catholics in the Diocese of Kansas City-St. Joseph taking this insult? If they have no care for survivors, do they at least wonder what kind of priests the seminarians who are ordained by Bishop Finn – and who uttered not one word of protest about it – will become in the lifetimes of priesthood ahead of them? Is that the kind of priest they are seeking as leaders of their parishes?
We honor the survivors and their families who will once again bear the cruelty of the Church's callousness.
The Church's tin ear in regard to this event is downright appalling but tin ears can be repaired – hardened hearts and hubris prove all the more telling.
— Kristine Ward, Chair, National Survivor Advocates Coalition (NSAC) Kristineward@hotmail.com
TEXT of the LETTER SENT TO BISHOPS by NSAC:
May 15, 2015 Dear Excellency: We sincerely request that you replace Bishop Robert Finn as the ordaining Bishop for the May 23, 2015 ordination of seminarians to the priesthood in the Diocese of Kansas City-St. Joseph. We understand Bishop Finn can perform the ordinations but we seek your assistance in preventing yet another blow to the survivors of sexual abuse by priests and members of religious orders of the Roman Catholic Church.
While another Bishop can replace Bishop Finn for this public ceremony, no one can replace a survivor. It is the survivors' pain that we believe should be a concern of the heart and soul of the Church as these ordinations approach and are carried out by a Bishop who was and remains a central figure in this crisis. There is no way healing can be real and true if actions such as Bishop Finn presiding at ordinations on May 23 following his resignation nearly three years after his conviction are allowed to continue and are viewed as good decisions.
The National Survivor Advocates Coalition (NSAC) would be happy to pay your expenses and arrange for your transportation if it would assist you in being the presiding Bishop at the ordination ceremony.
If you choose not to be the presiding Bishop we ask for your action to contact Bishop Joseph F. Naumann, Bishop of Diocese of Kansas City, Kansas and the temporary administrator of the Diocese of Kansas City- St. Joseph and with fraternal love and in the spirit of fraternal correction vigorously prevail upon him to change the time of the Diocese of Kansas City-St. Joseph ordinations in order for his schedule to permit him to be the ordaining bishop or that the Diocese of Kansas City-St. Joseph men who will be ordained priests be included in the ordination ceremony planned by his own diocese where he will be the presiding Bishop.
We appreciate your consideration of our requests.
Sincerely, Kristine Ward Chair, National Survivor Advocates Coalition
This entry was posted in Editorials and tagged Apostolic d, Apostolic Delegate to the United States, Archbishop Vigano, Bishop Naumann, Bishop Robert Finn, Cardinal Marc Ouellet, Diocese of Kansas City-St. Joseph, resignation of Bishop Finn on May 22, 2015 by nsac1.
We read with astonishment that Bishop Robert Finn will preside at the ordinations of seven seminaries in the Kansas City- St. Joseph Diocese in May because Archbishop Joseph F. Naumann who temporarily has charge of this diocese has a scheduling conflict.
Tone deaf is a profound understatement on this matter.
We understand that Bishop Finn has the power to ordain these seminarians regardless of his "voluntary" resignation that was accepted by Pope Francis.
But what in heaven's name makes anyone in the power structure or the pews of the Roman Catholic Church think this is acceptable?
This is about as acceptable as Richard Nixon presiding over a Cabinet meeting for Gerald Ford or signing an executive order because Ford had some work on Capitol Hill that he had previously scheduled.
We get that Archbishop Naumann already has ordinations scheduled. 'Tis the season for them.
But how about the hierarchy of the Roman Catholic Church – and those who are being ordained – act in the same way real people do. When a major event has a major problem, maturity goes into action and a solution is found that doesn't insult anybody.
How about the time of the ordination be moved to later in the day?
How about moving the ordination to the next day or the next weekend or the next month? We understand that the seminarians and their families and friends have planned for this event and it will cause some difficulty if the event were moved even by a day, but plenty of people make sacrifices to be at events of this significance for the people they love. And, really, we think the family members of the seminarians would be all too willing to make whatever adjustments were necessary if an ordination date had to be moved. It's not for them that this accommodation of Finn's presiding is being made. And certainly the seminarians should not be adverse to a bit of sacrifice on their way to ordination.
How about everybody understand why Bishop Finn "voluntarily" resigned and that should mean he won't be available to perform this highly public act of power and authority within the Church.
According to the United States Conference of Catholic Bishops (USCCB) there are 446 active and retired bishops in the United States – all of them have the power to ordain.
Will one of the other 445 please step forward and remove this insult to survivors, to people in the pews, to men and women of goodwill who cannot understand why it took nearly three years to remove Bishop Finn, and now, in cavalier fashion, we come to this announcement because of a scheduling conflict?
The National Survivor Advocates Coalition (NSAC) will pick up the transportation costs if the airfare, gas money or train ticket is what is keeping another bishop from raising his hand to take over this ordination. Enough is positively enough!
And the seminarians should have some say in this. If they think it's okay, maybe a hard look should be taken at them to see whom they plan to follow as a priestly model.
In case you missed the news story in NSAC News, here's a link: http://www.kansascity.com/news/local/article19762005.html
— Kristine Ward, Chair, National Survivor Advocates Coalition (NSAC), KristineWard@hotmail.com
This entry was posted in Editorials and tagged Archbisop Naumann, Bishop Finn, Finn resignation on April 29, 2015 by nsac1.
The Voluntary Resignation
Voluntary resignation instead of firing is not justice.
It is part of the dance of a gentlemen's game.
Yes, Bishop Robert Finn would no longer be the head of the Diocese of Kansas City- St. Joseph under either circumstance but it is important to make the distinction.
The permitting of a voluntary resignation allows Finn to retain the financial support that any retired bishop receives from the Roman Catholic Church and gives him a glide path of exit.
With the resignation coming at age 62, — albeit two-plus years beyond when it should have come and been accepted, – that leaves a lot of years for financial support to continue. That support comes from collection baskets, make no mistake about it.
We'd like to hear in-the-pew Catholics raise a bit of noise about this. Many pew occupiers had and have no difficulty slinging arrows of castigation that frame victims of molestation by priests and nuns as money grubbers.
For justice, the survivors with great courage went/go to court against a Church that had and has no problem lawyering up. Victims have been put through a second torture in having to testify about the molestation they have suffered.
Reparations for survivors are not even discussed within the broad forums of the Church.
We don't think Finn should be allowed a gracious retirement free of any obligation to serve. There are plenty of soup kitchens, food pantries, homeless shelters, and other activities in the Church where Catholics without episcopal rank do the work of the Gospel every day. Robert Finn should be shoulder to shoulder with them.
We don't think Finn should be allowed to leave Kansas City to find a comfortable life without financial worry.
After all, the survivors are condemned to a world of pain, scars, financial uncertainty, deprivation, mental instability, PTSD, broken relationships, isolation, physical wounds and other horrors every day.
Pope Francis, at the very least, should make clear that a voluntary resignation was a gift given to Finn and he should, as BishopAccountability.org requested yesterday, make clear that Finn's departure is the result of what he failed to do to protect children in this crisis.
Seeking to clear protestors from the streets of the US cities where Pope Francis will visit in September is undoubtedly part of the Vatican's double play of the whimper end of the investigation of religious orders of women and the removal of Finn. Do the Chileans have to wait until Pope Francis schedules a trip there to be rid of Bishop Barros?
The National Survivor Advocates Coalition (NSAC) said in its reaction statement yesterday to the news of the removal of Finn, which, semantics aside, is indeed what it was:
It's about time. The resignation of Bishop Finn as head of the Diocese of Kansas City- St. Joseph is not a moment for applause in the continuing crisis of sexual abuse in the Roman Catholic Church. It is a moment to ask why it took so long. It is a moment to ask when the others who protected and continue to protect abusers will be removed. It is a moment to ask why there are continuing defenders of bishops and religious superiors for predator priests and religious men and women, and diocesan staffs. May today's announcement, terse as it was from the Vatican, give a measure of peace to the survivors.
We restate it today with another call for justice and with a hope that those who have been blind and deaf to what has been going in the Church will see and hear.
We restate it today to emphasize that it is now clear that no papal commission needed to be created for papal action to be taken to remove a hierarch who protected an abuser.
We restate it today to lift our voices in a hearty call for all other hierarchs and religious superiors of men and women religious who have done what Robert Finn did – protect an abuser – voluntarily submit their resignations to Pope Francis.
Kristine Ward, Chair, National Survivor Advocates Coalition (NSAC) Kristineward@hotmail.com
This entry was posted in Editorials and tagged Bishop Robert Finn, Diocese of Kansas City-St. Joseph, Pope Francis, resignation of Bishop Finn on April 23, 2015 by nsac1.
Bishop Finn
National Survivor Advocates Coalition (NSAC) Statement on the Acceptance of Bishop Robert Finn's Resignation by Pope Francis
For Immediate Release – April 21. 2015
Contact: Kristine Ward, Chair, National Survivor Advocates Coalition (NSAC) Kristineward@hotmail. com, 937-272-0308
The resignation of Bishop Finn as head of the Diocese of Kansas City- St. Joseph is not a moment for applause in the continuing crisis of sexual abuse in the Roman Catholic Church.
It is a moment to ask why it took so long.
It is a moment to ask when the others who protected and continue to protect abusers will be removed.
It is a moment to ask why there are continuing defenders of bishops and religious superiors for predator priests and religious men and women, and diocesan staffs.
May today's announcement, terse as it was from the Vatican, give a measure of peace to the survivors.
— Kristine Ward, Chair, National Survivor Advocates Coalition (NSAC)
This entry was posted in Editorials and tagged Diocese of Kansas City-St. Joseph, finn, Pope Francis, sexual abuse, survivors, Vatican on April 21, 2015 by nsac1.
The Fixer
The National Survivor Advocates Coalition (NSAC) applauds the efforts of the two survivors on the papal commission regarding sexual abuse, Marie Collins of Ireland and Peter Saunders of the United Kingdom along with the two other commission members who supported their efforts to raise the issue at the Vatican of the unacceptability — to put it mildly — of Bishop Barros' appointment and installation as the head of the Diocese of Osorno, Chile.
Survivors in Chile accuse Bishop Juan de la Cruz Barros Madrid of covering up for the Rev. Fernando Karadima whom the Vatican convicted of abuse in 20011. Three Chilean survivors accuse Barros of witnessing their abuse by Karadima
While we applaud the actions of the commission members, we are appalled that they are the people who had to initiate the action to speak with the commission chair, Cardinal Sean O'Malley and through him seek to get the attention of Pope Francis.
Cardinal O'Malley, who not only is president of the papal sexual abuse comission but is a member of the Pope's kitchen cabinet, formally known as the Council of Nine, gets a newspaper and has access to radio, television, and Internet – and as a blogging Cardinal he certainly knows how to use the technology of communications.
He is not unaware by a longshot of the protests and objections to Bishop Barros and the reasons why this appointment needs to be rescinded.
It is Cardinal O'Malley who should have assured the members of the papal sexual abuse commission that he had already contacted Pope Francis and the issue would be front and center on this week's meeting of the Council of Nine with Pope Francis, eliminating any need for the survivors and the other two commission members to travel to Rome or entreat him to seek redress of this appointment that insults and hurts survivors.
But Cardinal O'Malley didn't take that action. He continued his long history as a fixer for the Vatican regarding sexual abuse beginning with his appointment in 1992 as the bishop of Fall River, MA as a clean up action of the revelations of the notorious abuser James Porter.
Sitting at a meeting with survivors and agreeing to convey their sentiments to another authority is a hallmark of Cardinal O'Malley's. He has finely honed the skill of the appearance of action and empathy.
If you enjoy theater, Cardinal O'Malley's performance that builds yet another protective tent for hierarchs while continuing to disguise him as a champion of reform is stellar.
The news reports of the meeting that the commission members held with Cardinal O'Malley on Sunday evening reveal that all Cardinal O'Malley agreed to do was raise the issue with Pope Francis. An issue, mind you, that the commission members had to bring to him, not one that he could see beforehand.
We have no doubt that Cardinal O'Malley was true to his word. He brought the issue up at a session of the Council of Nine with Pope Francis present. What that did was result in the speaking about what the Pope and Cardinals already know — there is opposition to the appointment of Bishop Barros by survivors and members of the papal abuse commission.
ZENIT's report of the Council of Nine's meeting includes these paragraphs http://www.zenit.org/en (Ninth Meeting of the Council of Nine Concludes):
During the meeting of the Council of Cardinals, Cardinal Sean O'Malley, president of the Commission for the Protection of Minors, discussed the theme of accountability regarding superiors (either bishops or religious) of priests involved in cases of abuse.
The Council went on to evaluate the methods and procedures in confronting the "failure of responsibility" and "abuse of office" on the part of authorities responsible for supervising.
According to Fr. Lombardi, Cardinal O'Malley met on Sunday with members of the Commission for the Protection of Minors, who asked him to "make present to the Pope their worries on those who have important roles in the Church and their qualifications for assuming adequate responsibility for the protection of minors."
A "precise and reliable legal text" will thus be assessed that defines the responsibilities of superiors, Fr. Lombardi said.
What Cardinal O'Malley has done is open a huge umbrella of protection for Bishop Barros.
Cardinal O'Malley, as he is expert at doing, has pushed the ephemeral, ever promised "accountability", once again into that beautiful future that is always described and tantalizingly almost within reach — but never arrives.
in the wonderful future constructed by Cardinal O'Malley to protect his brother bishops while looking like he is taking up the cause of survivors, Barros will not have failed at responsibility. His "failure" will have occurred before he was appointed as a bishop when he had no "responsibility" — outside of a moral one but, hey, he was only a priest not a bishop.
No "precise and reliable legal text" that may or may not come about in the future will touch him — nor any of his colleague bishops who have aided and abetted the rape and sodomy of children by protecting the clerics and religious men and women who molested them.
And no one is promising that any such "reliable and legal text" will be adopted, only as the statement says that it will be "assessed." Cue the mack truck, please.
If Pope Francis wanted to hear the survivors he could have met with them in the Saint Martha guesthouse or any place in the Vatican. If he wanted to talk with the Chilean victims he could.
If Cardinal O'Malley really wanted to make a difference he could. After all, he has a telephone and we know Pope Francis has one. So does Cardinal Marc Ouellet who heads the Congregation for Bishops.
This issue could have been resolved before the appointment was made, at the time the survivors made known their views in news stories, when the interruption of Bishop Barros' installation ceremony took place, when the numerous letters and contacts were made to Pope Francis, and before the Vatican issued the statement of the Congregation for Bishops backing the already-made appointment.
Whenever a ranking prelate wants to quash the appointment of a cleric to a bishopric it happens — let alone when a pope objects.
We hope that all men and women of goodwill remember that bishops become bishops and remain as bishops because of the actions of popes — including very popular and charming popes.
There is much more at work in this appointment. The influence of Cardinal Angelo Sodano, who has had overbearing influence in Chilean matters, cannot be discounted.
Cardinal O'Malley knows that.
He also knows that Cardinal Ouellet is sitting as the Prefect of the Congregation of Bishops because of the number of votes he received for pope in the last conclave.
Cardinal O'Malley also knows that it borders on the near-impossible that Bishop Barros' appointment will be rescinded.
Now, the only face-saving action for Pope Francis and the Vatican and Bishop Barros is for Bishop Barros to — or appear to — voluntarily resign.
If he wanted to, Cardinal O'Malley could be a fixer for that.
—- Kristine Ward, Chair, National Survivor Advocates Coalition (NSAC)
This entry was posted in Editorials and tagged Barros, Cardinal O'Malley, Cardinal Sean O'Malley, Chilean bishop, Council, Council of Nine, Karadima, Pope Francis on April 17, 2015 by nsac1.
Why Have You Forsaken Them?
The question has to be asked.
It has to be asked of Pope Francis.
There is no better week to ask it than Holy Week.
No better words than those of the Lord's.
Words wrung out in the ebbing away of life. Words labored and laden with anguish, pain, and abandonment.
It has to be the place where the Chilean victims and by extension all survivors of sexual abuse by priests and religious are today in the face of the brutal slap they have received from Pope Francis' backing of the declaration of the Congregation for Bishops supporting Juan de la Cruz Barros Madrid as the newly appointed and installed bishop of Osorno, Chile.
This is the Chilean bishop whose installation was interrupted by loud protestations within the cathedral. And how often has any Catholic seen that happen?
The Vatican's press statement said this:
Prior to the recent appointment of His Excellency Msgr. Juan de la Cruz Barros Madrid as bishop of Osorno, Chile, the Congregation for Bishops carefully examined the prelate's candidature and did not find objective reasons to preclude the appointment.
For those who cannot bring themselves to think that Pope Francis does not back this appointment, please read on from the report in ZENIT:
Fr. (Ciro) Benedettini's (Vice Director of the Holy See Press Office) statement echos those of Archbishop Fernando Chomali Garib of Concepcion, who in an interview with Chile's El Sur newspaper on March 26th, stated that he spoke personally with Pope Francis regarding the appointment.
"All the documentation that I cited came to him (Pope Francis), whether through the nunciature or the Chilean Embassy to the Holy See. He was very much up to date on Bishop Barros' situation, and in fact, a few days prior he had spoken with him," Archbishop Chomali said.
"With firmness and much conviction, he told me that he had analyzed all the past records and that there was no objective reason that Bishop Barros should not be installed as diocesan bishop."
Here is the link to the ZENIT news story on this issue: http://www.zenit.org/en
The Congregation's action is not surprising, sad but not surprising – and given the history of the public revelations of the sexual abuse crisis, it could have been anticipated: this band of brothers hangs together.
But for this Father, the Holy Father of this band of brothers to have heavy lidded eyes through which mercy cannot penetrate is a stinging whip or rebuke to survivors.
Is this truly the only person who could be the Bishop of Osorno, Chile?
The Chilean survivors have told the world that Bishop Barros was a witness to their abuse by Fr. Fernando Karadima.
Survivors carry enough burden in their lives. Abusers and their protectors are plentiful in number and there is no need to invent them or create a spectacle in a cathedral as a headline grabbing flashpoint overtaken soon enough by the next spectacle. Does anyone really think these protestors had nothing else to do that day?
Where is the concern of this bishop for the survivors that he would not lay down this position of bishop or not accept it in the first place?
Who are the poor?
What does a Jubilee Year of Mercy mean if not to overlook and overcome the self and seek the good of the other and seek it in an overflowing abundance of love?
In yet another ZENIT news story, we read that Pope Francis has been named by Fortune magazine as the fourth "greatest world leader."
The ranking are given, Fortune says to "extraordinary men and women who are transforming business, government, philanthropy, and so much more. "
Fortune said Pope Francis "has been shaking up the management of one of the world's largest bureaucracies: the Roman Catholic Church."
"It is not just that he has led by example—by now it's well known that the pope, who has long championed the virtues of charity and modesty, has forgone the traditional suite in the Apostolic Palace, opting instead to reside in a one-bedroom apartment in the Vatican guesthouse."
"Less known is how decisive he is in personnel choices," the magazine continued, "replacing the boards of the Vatican Bank and its main regulatory body with highly respected business-people from around the globe."
Despite perhaps some pushback, "this pontiff is not easily conned," Fortune says. "This, after all, is a pope who lives his own lessons."
The personnel choice here is a bad one.
The rubber is hitting the road and the smell is not sweet.
It is important to be clear here by making and letting this appointment of Bishop Barros stand Pope Francis is declaring that he is not going to take action against those who aided and abetted abusers. He is going to protect them.
We have Bishop Robert Finn of the Diocese of Kansas City- St. Joseph as a glaring example of that.
Now, Bishop Barros.
The problem of sexual abuse needs no other hand for its correction than Pope Francis'.
This appointment gives the back of it to survivors.
With all the power needed and necessary residing in him, the question has to be asked: Why has Pope Francis, who so clearly hears the cry of the poor, the innocent, the vulnerable and the marginalized, forsaken the survivors – the very children who were striped naked, brutalized, raped and sodomized by the agents of the Church over whom he has absolute power?
—- Kristine Ward, Chair National Survivor Advocates Coalition (NSAC) KristineWard@hotmail.com
This entry was posted in Editorials and tagged Archbishop Chomali, Bishop Finn, Bishop of Osorno, Chile, Fr. Fernando Karadima., Juan de la Cruz Barros Madrid, Kristine Ward, Pope Francis, sex scandal in Chile on April 2, 2015 by nsac1.
Un-hatting, Anyone?
The world knows Pope Francis can do surprising things.
He is not hide bound to tradition. The evidence is clear on a number of fronts, the latest of which being his Sunday announcement of the second round of cardinal appointments of his papacy.
Now, how about a surprise that might shake some sense into his hierarchy: un-hatting.
Today, January 6, 2015, — thirteen years into the Boston revelations of the sexual abuse scandal that became the tidal wave of revelations throughout the United States and rippling, then flooding across the world, — seems a most fitting day to consider un-hatting.
Cardinals who don't deserve their station should have their red hats rescinded. If accountability is a real end of Francis the Reformer, then Francis the Giver should also be Francis the Taker Awayer – and no hat should come with a lifetime guarantee.
If the affront of a naked head would be too much to bear for those who made deliberate decisions not to protect children, then let Pope Francis create a switching-out day as well as a cardinal-making day. Switching out day could swap red hats for black ones.
Good Friday seems a fitting day for that — without a third day return policy built in, of course.
If not Good Friday, then any day that is a cause for fireworks. Or how about Chinese New Year which this year falls on February 19, just five days beyond the hatting of Francis' new crop of cardinals. And interestingly enough, this year it's the Year of the Sheep.
Un-hatting for the dereliction of duty which allowed and then turned a blind eye to the rape and sodomy of children seems a balanced opposite of what cardinals hear they are supposed to live up to when they kneel before a pope and receive a red hat:
"To the glory of Almighty God and the honour of the Apostolic See, receive the scarlet biretta as a sign of the dignity of the cardinalate, signifying your readiness to act with courage, even to the shedding of your blood, for the increase of the Christian faith, for the peace and tranquility of the people of God and for the freedom and growth of Holy Roman Church."
Indeed, it must be said that for what is passed off as courage by most cardinals when juxtaposed to those in the world who daily are under the threat of the shedding of their blood, these words are betrayed by pomposity and eviscerated of the nobility of their promise.
What peace and tranquility is promoted when victims and their families are rent asunder by the savageness of the physical and psychological wounds of sexual abuse?
Who are the poor, if not the victims of sexual abuse?
Who are human trafficking victims, if not these?
Who are those whose freedom was unmercifully, selfishly, cruelly taken, if not these?
Pope Francis may say as many times as he wishes that being named a cardinal is not a promotion, but the perception of reality is reality and, make no mistake, news stories, conversation, comment and convention regarding these men will carry the words "elevated, prince, ceremony , celebration and exclusivity and privilege."
Thirteen long years into a major revelations of the sexual abuse scandal in the United States, not to mention the revelations from Ireland, the United Kingdom, Belgium, Poland, Australia, New Zealand, Chile, and Argentina, to mention a few — and three popes on — we've got a papal commission peopled by only two survivors.
Surely that does not measure up to anyone's definition of courage.
If the singular role of cardinals in the Church is the election of a pope can we really not find better people than these among more than a billion Catholics to elect the Church's leader? Really?
This list of the un-hatted should begin with Bernard Law and Roger Mahony but certainly should not end with them — and it should not end within the ranks of cardinals. The list of bishops that should be un-hatted should begin with Robert Finn.
Isn't it time for a real epiphany?
Think anew, St. Paul tells us.
Let the un-hatting begin.
This entry was posted in Editorials on January 6, 2015 by nsac1.
Finn: A Fit Leader?
The National Catholic Reporter (NCR) broke a story this week that Archbishop Terrence Prendergast of Ottawa has conducted an apostolic visitation — at the behest of the Vatican — of the Diocese of Kansas City-St. Joseph, Missouri.
The Archbishop, it is reported, asked of supporters and defenders of Bishop Robert Finn: Do you think Bishop Finn is fit to be a leader?
It is possible that given all of the mounds of heart breaking evidence that's been produced through the courage of the survivors in this crisis – evidence that has been ignored, attempted to be explained away, and pushed under massive rugs – Oriental and otherwise — and vigorously assigned – but not staying put – as history, — that the Archbishop will answer his question in the affirmative and deliver the answer to the Vatican that Finn is a fit leader.
Or he if answers no that the Vatican and Pope Francis himself may decide that Finn should remain in place.
But, while we still are that fluid zone where the decision is yet to be make – or, if made, not yet carried out – it may be possible to entertain the thought of what hell freezing over would sound and look like.
A cautionary damper is placed on this, though, by the at first hopeful news and then disclaimered news that the reason Paraguayan Bishop Rogelio Livieres Plano was removed from office was his protection of Monsignor Carolos Urritigoity against whom a federal lawsuit charging abuse was filed in the United States. The suit was followed by a bishop to bishop warning from Diocese of Scranton Bishop James Timlin not to give Utttigoity faculties in Paraguay.
Now in the case of Finn's fitness, Archbishop Prendergast stands in the breach.
We urge any one with information regarding Bishop Finn's and his diocese's conduct in regard to the Shawn Ratigan child pornography case and the Diocese's protestation of payment of $1.1 million ordered by an arbitrator for violating the 2008 settlement conditions for victims of sexual abuse by priests in the diocese to raise their voices, their pen, their texts, their email, their faxes – and any other method of preferred communication.
We mean everyone with information: survivors, family members of survivors, neighbors, friends, classmates, teammates of survivors, attorneys, advocates, people in the pews, concerned citizens, chancery officials, priests, deacons, religious, and brother bishops.
We urge them to answer Archbishop Prendergast's question: is Finn fit to be a leader?
Here is Archbishop Prendergast's contact information through the Archdiocese of Ottawa:
The Diocesan Centre
1247 Kilborn Place Ottawa, Ontario
K1H 6K9
Telephone number: 613-738-5025 The voicemail system, the Archdiocese says, is available 24/7 Fax number: 613-738-0130 E-mail: reception@archottawa.ca
Or send your information to us and we'll forward it.
We know that the Catholics of the Diocese of Kansas City-St. Joseph deserve better.
We certainly know the children who live in the Diocese of Kansas City- St. Joseph- Catholic or not – deserve better.
We know that it is incredible to believe that among the 38,275 priests in the United States there is not to be found a more fit leader for this Diocese.
We know that Bishop Finn should have found it within himself to offer his resignation.
The issue parallels the director of the Secret Service resignation. The logic is the same: when the persons you and your organization are charged to protect come into and remain in harm's way because of the action, inaction, policies, inattention or incompetence, of your organization the "leader" needs to step aside for the integrity of the organization. The realization of what's needs doing should not be a drawn out process. .
There is sensibility in resignation. Those who believe themselves to be leaders and those who keep them in positions of leadership should be able to see this. And not only this, but the horizon of harm that it does to the Church and the society in general when this kind of "leadership" is allowed to go on.
No waiting game should be played for a polite exit for Bishop Finn. No age game, no naming of a co-adjutor auxiliary (although for a diocese so small this would be a stretch), no walking away from plain truth.
We know that one can serve a Church without being a bishop and that being named one should not come with an unexamined lifetime guarantee.
We know it should be evident to Pope Francis – as it should have been to Pope Benedict — that a bishop who has conducted himself as Bishop Finn has is not fit to be a leader.
If it is, it is a new low in the standards of what should be sought in a candidate for bishop.
We also know that silence is not an option.
What's at stake here ultimately are children.
— Kristine Ward, Chair, National Survivor Advocates Coalition, (NSAC) KristineWard@hotmail.com
This entry was posted in Editorials and tagged Archbishop Ottawa, Archbishop Prendergast, Paraguayan Bishop Rogelio Livieres Plano on October 3, 2014 by nsac1.
The Choice for Chicago
Pope Francis' choice to replace Cardinal Francis George is Bishop Blase Cupich who will leave the Diocese of Spokane, Washington, to take over the helm at the Archdiocese of Chicago.
Apropos of nature's season of changing colors, Cupich's official outerwear will undoubtedly change from magenta to red in the not too distant future, again significantly ratcheting up his influence within the Church's internal structure.
There will be mountains of words written about this choice now, at his installation, and at the time of the change-of-color events.
For now, deep into this crisis of monumental proportions with a huge Vatican public relations operation underway, we think the proof still remains in the pudding.
No matter what descriptive label is affixed to this choice, or parallel peering into the motivations of Pope Francis, or turn in the road that may be imagined or real, we think it suffices to say only these two words: we'll see.
* Kristine Ward, Chair, National Survivor Advocates Coalition, Kristineward@hotmail.com
This entry was posted in Editorials on September 22, 2014 by nsac1.
Action Alert (1)
Kristine Ward (22)
Bilgrimage
Bishop-Accountability website
CNN Belief Blog
Kris Ward Dayton TV
National Catholic Reporter
NSAC's New Website
Patrick Wall's blog
Progressive Catholic Voice

redpajama_set_name: RedPajamaCommonCrawl


text:
When Ronaldinho Scored A Goal So Good The Real Madrid Fans Applauded Him
Jack Kenmare
Published 16:59, 19 March 2022 GMT
| Last updated 11:42, 04 May 2022 BST
It remains one of the most talked about moments in El Clasico history. The day Ronaldinho brought The Bernabeu to its feet.
On November 19, 2005, thousands of fans inside the Bernabeu applauded as a helpless Iker Casillas shrugged his shoulders, powerless to the genius of a Brazilian-born magician.
It was a rare glimpse of admiration from the often indignant Los Blancos support. Check out that memorable game below.
Ronaldinho's marvelous efforts just had to be appreciated from the masses. He was truly unplayable that night, ducking and weaving through players like never before.
The 25-year-old scored two superb individual goals as the Catalan giants romped to victory over their biggest rivals, and it all began on the hour mark, when the Barca man skipped past two challenges with ease and fired past Casillas.
Los Blancos looked on helpless. In fact, they looked on in fear, trying to guess 'Dinho's next move.
Ronaldinho Gaucho masterclass for Barcelona vs Real Madrid in the #ElClasico
Sergio Ramos saw things that day!
Real Madrid fans applauded him 👏pic.twitter.com/9A0JUoKLLT
— Elijah Kyama (@ElijahKyama) March 18, 2022
They were right to be wary. Madrid's defence crumbled as Ronaldinho ran towards them.
In a similar turn of events, the Barcelona man took on two defenders on the left to slot home past. 3-0.
It was a moment to cherish for everyone in attendance. Iconically, a moustached Real Madrid fan, who wore a scarf over his neck, stood up and started cheering what he just had witnessed.
Then, in a surprise move, everyone else joined in.
Image: La Liga
Samuel Eto'o may have put the visitors ahead just after 15 minutes but his performance dissolved into the unknown, perilous to the Ronaldinho show.
Shortly after the game, Eto'o spoke to Barça TV and his reaction said it all.
"When Ronaldinho scored that goal when he got rid of everyone, even his teammates, I stood there in disbelief, looked to one side and muttered to myself: 'But, what is happening?' It was unbelievable." he said.
Ronaldinho went on to admit that he didn't even realise he was being applauded by the Real Madrid support:
"No, I didn't realise in the moment." he said. "I remembered when we arrived to the locker room and everybody was talking about it and I went to check it on the TV.
"Because of the joy of scoring and the celebration, I didn't notice the standing ovation. It's special. Only a few players have had this joy of being acclaimed by the fans of your arch rivals in a derby."
Featured Image Credit: La Liga
Topics: Ronaldinho, Barcelona, Real Madrid, El Clasico
17 years ago today, Ronaldinho scored a goal so good Real Madrid fans couldn't help but applaud
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Tottenham Hotspur complete first direct signing from Chelsea in 14 years
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Barcelona are considering a shock move for Man Utd forward Anthony Elanga


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Maui Election
CANDIDATE SPOTLIGHT: Alana Kay – Candidate Maui Mayor
July 31, 2014, 12:02 AM HST
* Updated July 28, 7:23 AM
Alana Kay.
By Maui Now Staff
Alana Kay is a candidate for the office of Mayor in Maui County. Kay is among a list of 6 individuals identified as candidates in the race on the 2014 ballot.
The complete list of candidates (in alphabetical order) includes: Alan M Arakawa; Beau E Hawkes; Alana Kay; Orion "Ori" Kopelman; Tamara "Tam" Paltin; and Nelson Nahinu Waikiki Jr.
Full Name: Alana Kay
Date and location of Birth: December 19, 1959 in Milwaukee, Wisconsin
Current residence: 555 Kaukahi Street, Kīhei, HI 96753
Occupation: Self-employed author, publisher, Violet Phoenix Publishing
Elected offices held, if any: (blank)
Education: Working on Sustainable Science Management degree at UHMC
Community Involvement: Girl Scout Leader, Junior Achievement Advisor; Underground Switchboard, Crisis Line Counselor
Family status: Single, Mother of 3, Grandmother of 5
Maui Mayoral Candidate Questionnaire:
1. What is the number one issue facing Maui County? Why do you feel it warrants special attention and how do you propose to resolve this issue? If you have more than one item, please elaborate.
Answer: I believe that the number one issue facing Maui County is homelessness and affordable housing. At this time, a large number of Maui residents live below the sustenance level as well as the poverty level. An overwhelming percentage of Maui residents pay 50 – 70% of their gross income on housing, which makes them house-poor. At this time, it is estimated that there are anywhere from 3 to 10 thousand homeless in Maui County. This does not include those who have doubled up in housing. Those in the community who serve the homeless also believe that we have reached crisis level with regard to this issue. We need to begin by treating the under-employed and homeless with dignity. We need to direct a much larger portion of our budget and personnel toward homeless resources. We need to improve intake practices and create an innovative job matching website. Some homeless advocates have suggested purchasing the Haggai Institute building on Līpoa to be used as a resource center and temporary housing. Other municipalities and countries have developed high functioning homeless transition programs and we would be wise to study them and learn from them and also adapt our own strategies by listening to input from the brilliant minds in our community. Other than homelessness, Native Hawaiian issues and water distribution are high on the list.
2. What areas can be trimmed to make for a more efficient government; and what areas can afford additional funding to facilitate long-term benefits for the County?
Answer: We could save money by bringing our recycling and waste management programs into the 21st century. We would also create more cash flow for the Department of Water Supply if we would change the way we do business with HC&S and Wailuku Water Company. More money could then be put into water delivery infrastructure to improve availability and affordability for all residents, especially farmers. We also need to eliminate the injection wells and invest money into building new water treatment plants instead. This way, we would discontinue the polluting of the oceans and aquifer systems and enjoy a vastly more expedient way of processing our water for re-use. The future of Maui is food independence and sustainable practices, so all efforts to enhance farming and eco friendly practices would be an investment in our future.
***Candidate spotlight segments were compiled as a public service to further educate and inform Maui voters about the upcoming primary election on August 9, 2014, and the general election on November 4, 2014. The questionnaires were distributed via mail following the primary election candidate filing deadline for select races affecting Maui County. The responses do not reflect the opinion or position of Maui Now or Pacific Media Group.
***Note: The name of voided Maui mayoral candidate Neldon E "ADZ" Mamuad will still appear on the primary election ballot, but votes for him will not be counted toward the end result.
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Hoaloha Farms in The Business Spotlight January 31, 2019


As of Friday morning, nearly 600,000 people had signed a petition calling on the members of the Electoral College to stage a coup on December 19, vote for Hillary Clinton regardless of their state's outcome, and prevent Donald Trump from becoming the 45th President of the United States. In other words, hundreds of thousands of liberals are ready to turn talk of a rigged election into a reality.
Mr. Trump is unfit to serve. His scapegoating of so many Americans, and his impulsivity, bullying, lying, admitted history of sexual assault and utter lack of experience make him a danger to the Republic.
It's hard to take seriously those who are now pushing for the abolition of the Electoral College. Whether or not that's a discussion we should have, your argument loses all credibility when you only make it after losing. Bring it up when your side wins the Electoral College and loses the popular vote, and maybe you can get people to listen. Talking about it now just sounds like sour grapes.
And that doesn't even get into the scheme they're seriously proposing here. Have any of these people actually considered what would happen if their wild dream came true? Do they imagine that 60 million Trump voters would just shrug if the Electoral College handed the election to Hillary Clinton? What they're proposing is the wholesale theft of the election, an event that would put us closer to an actual civil war than we've been in 150 years.
It brings you back to May, when certain Republicans wanted the RNC to ignore the outcome of the primaries and find a way to give someone other than Trump the nomination. Thankfully, cooler heads prevailed. Thankfully, there were enough Republican officials who understood that even if they loathed Trump, they could not deliberately defy the voters and walk away unscathed. And even if they had pulled off such a coup, it would have been small potatoes compared to what liberals are proposing now. It would have been, at worst, the end of the RNC.
An Electoral College revolt would be – with no exaggeration whatsoever – the end of democracy as we know it.


Aodh Ruadh got their 2010 league campaign off to a good start when they took on a strong Malin side last Friday, 2nd April. In adverse weather conditions, it was a hard fought battle from beginning to end, with just two points in it at half time when the score line read, Aodh Ruadh 0-6 Malin 0-4. However in the second half Aodh Ruadh managed to hold Malin to just one point. The game finished 0-14 to 0-5. The team on the day was:Marita Maguire; Catherine Lloyd, Niamh Gallagher, Karol Foley; Grainne McCosker, Annette Lloyd, Katie Flood; Tanya O'Shea, Sarah Patton; Marita Leonard, Donna Martin, Emma McPhelim; Fionnuala Gill, Lorraine Roper, Donna Rogers. Andrea Rogers replaced Lorraine Roper who sustained an injury in the first half.