text:
\section{The topological reconstruction problem}

In 1941, Ulam and Kelly proposed the \emph{reconstruction conjecture} in graph theory. This conjecture asks whether every finite graph with at least three vertices is uniquely determined by its unlabelled subgraphs obtained by deleting a single vertex and all incident edges. To this day, the reconstruction conjecture has remained open, and is considered one of the most challenging problems in graph theory. For a survey about the reconstruction conjecture see for example \cite{Bondy}.

The present paper is concerned with the topological version of the reconstruction conjecture, introduced in \cite{recpaper}. A topological space $Y$ is called a \emph{card} of another space $X$ if $Y$ is homeomorphic to $X \setminus \Set{x}$ for some $x$ in $X$. The \emph{deck} $\singletonDeletion{X}$ of a space $X$ is a transversal for the non-homeomorphic cards of $X$, i.e.\ a set recording the topologically distinct subspaces one can obtain by deleting singletons from $X$. For example, if $Y$ is a card of the real line, then $Y$ is homeomorphic to two copies of the real line. We write this as $\singletonDeletion{\R}=\Set{\R \oplus \R}$. For $n$-dimensional Euclidean spaces we have $\singletonDeletion{\R^n}=\Set{\R^n \setminus \Set{0}}$. For the unit interval $I$ we have $\singletonDeletion{I}=\Set{[0,1), [0,1) \oplus [0,1)}$. The Cantor set has the deck $\singletonDeletion{C}=\Set{C \setminus \Set{0}}$ and the rationals $\Q$ and irrationals $P$ have decks $\singletonDeletion{\Q} = \Set{\Q}$ and $\singletonDeletion{P}=\Set{P}$.

Two topological spaces are said to be \emph{reconstructions} of each other if their decks are indistinguishable. A space $X$ is said to be \emph{reconstructible} if all its reconstructions are homeomorphic to $X$. In the same spirit, we say that a property of topological spaces is reconstructible if it is preserved under reconstruction. Formally, a space $X$ is reconstructible if $\singletonDeletion{X}=\singletonDeletion{Z}$ implies $X \cong Z$, and a property $\mathcal{P}$ of topological spaces is reconstructible if $\singletonDeletion{X}=\singletonDeletion{Z}$ implies ``$X$ has $\mathcal{P}$ if and only if $Z$ has $\mathcal{P}$".

It is shown in \cite{recpaper} that all the aforementioned examples are reconstructible, with the exception of the Cantor set where we have $\singletonDeletion{C}=\singletonDeletion{C\setminus \Set{0}}$. This example also shows that compactness is a non-reconstructible property. These observations give rise to the question what underlying principles make a topological space or a topological property reconstructible.

\begin{recproblem}
Determine which topological spaces and properties are reconstructible and which ones fail to be reconstructible.
\end{recproblem}

The purpose of this paper is to describe two surprising results about topological reconstruction. Our first result shows that it is undecidable in ZFC whether $\omega^*$, the Stone-\v{C}ech remainder of the integers, is reconstructible. Our second result is concerned with the question whether \emph{normality}, one of the fundamental topological separation properties, is reconstructible. We show that $\omega^*$ provides a natural example establishing that the answer is consistently negative.

The result about reconstruction of normality is somewhat curious, especially so because all other topological separation axioms are easily seen to be reconstructible for spaces containing at least three points \cite[Thm.~3.1]{recpaper}. In particular, the property of being a $T_1$, a Hausdorff or a Tychonoff space is reconstructible. But for normality, the best we could do in \cite{recpaper} was to show that normality of a space is reconstructible provided the space has at least one normal card. On the positive side, this result applies to every normal space containing a $G_\delta$-point, since normality is hereditary with respect to $F_\sigma$ subspaces \cite[2.1.E]{Eng}. In particular, normality is reconstructible in the realm of first-countable spaces. However, for an uncountable cardinal $\kappa$, the Cantor cube $2^\kappa$ provides an example of a compact Hausdorff space where all cards are non-normal. And given that the Cantor set $C \cong 2^\omega$ is non-reconstructible, one might suspect that these large Cantor cubes are examples witnessing that normality is non-reconstructible. However, $2^\kappa$ is reconstructible if $\kappa$ is uncountable \cite[2.1]{recpaper}.

This is where the Stone-\v{C}ech remainder of the integers $\omega^*$ enters the stage. It is a well-known open problem whether all cards of $\omega^*$ are non-normal (cf.\ \cite[Q13]{openproblems} and \cite{szymanski}). Under the Continuum Hypothesis (CH), however, it is a classical theorem independently due to Rajagopalan \cite{raj} and Warren \cite{Warren} that the answer is yes. Thus, under CH, the space $\omega^*$ is a further candidate to witness that normality might be non-reconstructible.

And indeed, we show below that $\omega^*$ provides a consistent example that normality is non-reconstructible. In our main result we prove that assuming CH, the space $\omega^* \setminus \Set{p}$ for a $P$-point $p$ of $\omega^*$ (under CH, this is a point $p$ with a nested neighbourhood base) is a non-normal reconstruction of the normal space $\omega^*$. Thus, CH implies that normality is non-reconstructible. We also show that CH (which is equivalent to $\omega_1=\omega_1^{<\omega_1}$) can be weakened to the assumption that there is an uncountable cardinal $\kappa$ with $\kappa = \kappa^{<\kappa}$. We will see that under this cardinal assumption, the spaces $S_\kappa$ defined by Negrepontis in \cite{negrepontis} provide further examples that normality is non-reconstructible. This ties in well with the previous non-reconstruction results, as the spaces $S_\kappa$ can be thought of simultaneously generalising the behaviour of the Cantor set $C$ and of $\omega^*$ under CH.

This paper is organised as follows. In Section \ref{section3}, we recall the relevant definitions and facts about the spaces $\omega^*$ and $S_\kappa$. In Section \ref{sectiondeck} we briefly describe the deck of $\omega^*$, and then turn to a closer investigation of cards of $\omega^*$ and $S_\kappa$ in Section \ref{sectionfincomp}. Our main technical result states that under CH, all finite compactifications of $\omega^* \setminus \Set{x}$ are homeomorphic to $\omega^*$ and, moreover, that all but at most one point added at infinity will be $P$-points.

In Section \ref{sectionnormality}, we recall why cards of $\omega^*$ fail to be normal under CH. We then prove that for uncountable $\kappa$ with $\kappa = \kappa^{<\kappa}$, all cards of $S_\kappa$ are non-normal. In Section \ref{section4}, we prove the announced reconstruction results. We show that under CH, the space $\omega^*$ is non-reconstructible, and more generally that under $\kappa = \kappa^{<\kappa}$, the space $S_\kappa$ is non-reconstructible. We also show that it follows from a theorem by van Douwen, Kunen and van Mill that it is consistent with MA$+\neg$CH that $\omega^*$ is reconstructible. Finally, in Section \ref{sectionalltogether} we combine our results from the earlier sections to show that the existence of an uncountable cardinal $\kappa$ with $\kappa = \kappa^{<\kappa}$ implies that normality is non-reconstructible. We conclude the paper in Section \ref{sectionquestions} with some questions.

I would like to thank my advisor Rolf Suabedissen for inspiring discussions about the topological reconstruction problem.

\section{\texorpdfstring{What we need to know about $\omega^*$ and $S_\kappa$}{Introduction to the Stone-Cech compactification of the integers}}
\label{section3}

In this section we recall crucial properties of the spaces $\omega^*$ and $S_\kappa$ which we use in the course of this paper. All this and more can be found in \cite{Ultrafilters,Rings, Intro}.

A subset $Y$ of a space $X$ is $C^*$-\emph{embedded} if every bounded real-valued continuous function on $Y$ can be extended to a continuous function on all of $X$. For a Tychonoff space $X$, we write $\beta X$ for its \emph{Stone-\v{C}ech compactification}, the unique compact Hausdorff space in which $X$ is densely $C^*$-embedded, and we write $X^*=\beta X \setminus X$ for the \emph{remainder} of $X$. A space is \emph{zero-dimensional} if it has a base of clopen (closed-and-open) sets.

A subset of a Tychonoff space of the form $f^{-1}(0)$ for some real-valued continuous function $f$ is called a \emph{zero-set}. A \emph{cozero-set} is the complement of a zero-set. A Tychonoff space $X$ is an $F$-\emph{space} if each cozero-set is $C^*$-embedded in $X$. A $G_\delta$ subset of $X$ is a subset which can be expressed as a countable intersection of open sets, and an $F_\sigma$ subset is a complement of a $G_\delta$, namely a set which can be expressed as a countable union of closed subsets. We shall need the following facts about $F$-spaces \cite[Ch.\,14]{Rings}.

\begin{enumerate}
\item A normal space is an $F$-space if and only if disjoint open $F_\sigma$-sets have disjoint closures.
\item Closed subspaces of normal $F$-spaces are again $F$-spaces.
\item Infinite closed subspaces of compact $F$-spaces contain a copy of $\beta \omega$. Therefore, compact $F$-spaces do not contain convergent sequences.
\end{enumerate}

The space $\omega^*$ is a compact zero-dimensional Hausdorff space of weight $\mathfrak{c}=2^{\aleph_0}$ without isolated points with the following extra properties: it is an $F$-space in which each non-empty $G_\delta$-set has non-empty interior. A space with these properties is also called \emph{Parovi\v{c}enko space}.

\begin{mythm}[Parovi\v{c}enko {\cite{parov}}, van Douwen and van Mill {\cite{douwenmill}}]
\label{parovtheorem}
CH is equivalent to the assertion that every Parovi\v{c}enko space is homeomorphic to $\omega^*$.
\end{mythm}

A $P$\emph{-point} is a point $p$ such that any countable intersection of neighbourhoods of $p$ contains again a neighbourhood of $p$. In other words, $p$ is a $P$-point if $p$ is in the interior of every $G_\delta$ containing $p$. The existence of $P$-points in $\omega^*$ was first proved as a consequence of the Continuum Hypothesis in \cite{Rudin}. The existence of $P$-points can also be shown under $MA+\neg CH$ \cite[2.5.5]{Intro}. On the other hand, it is consistent that $P$-points in $\omega^*$ do not exist \cite[2.7]{Intro}.  We list some facts about $P$-points in $\omega^*$ under CH.
\begin{enumerate}
\setcounter{enumi}{3}
\item Assuming [CH], a point $p \in \omega^*$ is a $P$-point if and only if $p$ has a nested neighbourhood base if and only if $p$ is not contained in the boundary of any open $F_\sigma$-set.
\item Assuming [CH], for every pair of $P$-points in $\omega^*$ there exists an autohomeomorphism of $\omega^*$ mapping one $P$-point to the other \cite{Rudin}.
\end{enumerate}

We now describe the spaces $S_\kappa$, which generalise the behaviour of $\omega^*$ under CH to higher cardinals $\kappa$. Note that in compact zero-dimensional spaces, cozero-sets are countable unions of clopen sets. This motivates the following definition from \cite[Ch.\,14]{Ultrafilters}. In a zero-dimensional space $X$, the ($X$-)\emph{type} of an open subset $U$ of $X$ is the least cardinal number $\tau=\tau(U)$ such that $U$ can be written as a union of $\tau$-many clopen subsets of $X$. A zero-dimensional space where open subsets of type less than $\kappa$ are $C^*$-embedded is called an $F_\kappa$-\emph{space}. In zero-dimensional compact spaces the notions of $F$- and $F_{\omega_1}$-space coincide.

\begin{enumerate}
\item[($1'$)] A normal space is an $F_\kappa$-space if and only if disjoint open sets of types less than $\kappa$ have disjoint closures \cite[6.5]{Ultrafilters}.
\end{enumerate}

Following \cite[1.2]{Dow}, we call a space a $\kappa$-\emph{Parovi\v{c}enko space} if it is a compact zero-dimensional $F_\kappa$-space of weight $\kappa^{<\kappa}$ without isolated points such that non-empty intersections of fewer than $\kappa$ many open sets have non-empty interior. The space $\omega^*$ is an ($\omega_1$-)Parovi\v{c}enko space.

\begin{mythm}[Negrepontis {\cite{negrepontis}}, Dow {\cite{Dow}}]
\label{negre}
The assumption $\kappa = \kappa^{<\kappa}$ is equivalent to the assertion that all $\kappa$-Parovi\v{c}enko spaces are homeomorphic.
\end{mythm}

If the condition $\kappa=\kappa^{<\kappa}$ is satisfied then the unique $\kappa$-Parovi\v{c}enko space exists and is denoted by $S_\kappa$ \cite[6.12]{Ultrafilters}. The space $S_\omega$ is homeomorphic to the Cantor space, and whenever the space $S_{\omega_1}$ exists, it is homeomorphic to $\omega^*$. The existence of uncountable cardinals satisfying the equality $\kappa=\kappa^{<\kappa}$ is independent of ZFC but an assumption like $\kappa^+=2^\kappa$ implies the equality for $\kappa^+$.

A $P_\kappa$\emph{-point} $p$ is a point such that the intersection of less than $\kappa$-many neighbourhoods of $p$ contains again an open neighbourhood of $p$. Thus, a  $P_{\omega_1}$-point is a $P$-point. For a proof that $S_\kappa$ contains $P_\kappa$-points see \cite[6.17]{Ultrafilters} or \cite[4.6]{comppaper}.
\begin{enumerate}
\item [($4'$)] Assume $\kappa=\kappa^{<\kappa}$. In $S_\kappa$, a point $p$ is a $P_\kappa$-point if and only if $p$ has a nested neighbourhood base if and only if $p$ is not contained in the boundary of any open set of type less than $\kappa$.
\item [($5'$)] Assume $\kappa=\kappa^{<\kappa}$. For every pair of $P_\kappa$-points in $S_\kappa$ there exists an autohomeomorphism of $S_\kappa$ mapping one $P_\kappa$-point to the other \cite[6.21]{Ultrafilters}.
\end{enumerate}

\section{\texorpdfstring{The deck of $\omega^*$}{The deck of w*}}
\label{sectiondeck}

We briefly describe the cards of $\omega^*$. Recall that the cards of a space correspond to the non-homeomorphic subspaces one can obtain by deleting singletons. It turns out that cards of compact Hausdorff spaces, and in particular cards of $\omega^*$ correspond to the different orbits under the action of its autohomeomorphism group.

A space X is \emph{homogeneous} if for every pair of points $x$ and $y$ of X there exists a homeomorphism of $X$ carrying $x$ to $y$. In general, we say $x$ and $y$ lie in the same \emph{orbit} of $X$ if $x$ can be mapped to $y$ by a homeomorphism of $X$. The orbits form equivalence classes and the collection of orbits is denoted by $X/_\sim$.

If $x$ and $y$ lie in the same orbit of $X$ then deleting either $x$ or $y$ gives the same card. For example, fact $(4)$ from the previous section yields that under CH, all cards of $\omega^*$ obtained by deleting a $P$-point are homeomorphic. Also, the deck of a homogeneous space consists of only one card, and $\cardinality{\singletonDeletion{X}} \leq \cardinality{X/_{\sim}}$ is true for any space $X$. We now show that for compact Hausdorff spaces, we have equality in the previous line.

\begin{mythm}
\label{homogeneous}
Let $X$ be a compact Hausdorff space. Then $\cardinality{\singletonDeletion{X}} = \cardinality{X/_{\sim}}$ and the different cards correspond bijectively to the orbits of $X$.
\end{mythm}

\begin{proof}
As $\cardinality{\singletonDeletion{X}} \leq \cardinality{X/_{\sim}}$ by the remark preceding the theorem, it is enough to prove that $\cardinality{\singletonDeletion{X}} \geq \cardinality{X/_{\sim}}$. If $X$ is homogeneous, the result is clear. If $X$ is not homogeneous, find $x$ and $y$ contained in different orbits. Suppose for a contradiction that the two cards obtained by deleting $x$ and $y$ are homeomorphic, i.e.\ that there exists a homeomorphism $f \colon X \setminus \Set{x} \to X \setminus \Set{y}$. If $x$ is isolated, then both $X \setminus \Set{x}$ and $X \setminus \Set{y}$ are compact and hence $y$ must be isolated, too. But then, both points $x$ and $y$ lie in the same orbit, a contradiction.

Thus, we may assume that both $x$ and $y$ are non-isolated. Then $X$ is a one-point compactification of both $X \setminus \Set{x}$ and $X \setminus \Set{y}$. But  since all one-point compactifications of a locally compact space are homeomorphic by a map carrying remainders onto remainders \cite[3.5.11]{Eng}, the map
$f \cup \Set{\langle x,y \rangle}$ is a homeomorphism of $X$. Thus, $x$ and $y$ are contained in the same orbit, a contradiction.
\end{proof}

Z. Frol\'ik proved (in ZFC) that every orbit in $\omega^*$ is of size $\mathfrak{c}$, and therefore that $\omega^*$ has $2^\mathfrak{c}$-many orbits \cite{Frolik}. It follows that $\cardinality{\singletonDeletion{\omega^*}}= 2^\mathfrak{c} = \cardinality{\omega^*}$, i.e.\ the space $\omega^*$ has the maximal possible number of different cards.

\section{\texorpdfstring{Finite compactifications of cards of $\omega^*$ and $S_\kappa$}{Finite compactifications}}
\label{sectionfincomp}

This section contains the technical groundwork for our non-reconstruction results. The main result is a characterisation of finite compactifications of cards of $\omega^*$ and $S_\kappa$.

We call a point $x$ of a Hausdorff space $X$ a \emph{strong butterfly point} if its complement $X \setminus \Set{x}$ can be partitioned into open sets $A$ and $B$ such that $\closure{A} \cap \closure{B} = \Set{x}$. The sets $A$ and $B$ are called \emph{wings} of the butterfly point $x$. In the following we recall a theorem by Fine and Gillman \cite{fine,Gillmann} stating that under [CH], every point in $\omega^*$ is a strong butterfly point.

The result plays a crucial role in our proofs of Theorem \ref{ClassificationCompactifications} on the classification of finite compactifications of cards of $\omega^*$ and in our proof of Theorem \ref{nonnormalcards}, where we show that cards of $\omega^*$ are non-normal under CH.

\begin{mythm}[Butterfly Lemma, Fine and Gillman]
\label{butterfly}
\textnormal{[CH].} Every point in $\omega^*$ is a strong butterfly point. \qed
\end{mythm}

\begin{mycor}
\label{nonStone}
\textnormal{[CH].} Every card of $\omega^*$ has a two-point compactification. In particular, no card $\omega^* \setminus \Set{x}$ is $C^*$-embedded in $\omega^*$.
\end{mycor}

\begin{proof}
It follows from the Butterfly Lemma that for every point $x$ in $\omega^*$, we have $\omega^* \setminus \Set{x} = A \oplus B$ where $A$ and $B$ are non-compact open subsets of $\omega^*$. Considering their respective one-point compactifications $\alpha A$ and $\alpha B$, we see that $\alpha A \oplus \alpha B$ is a two-point compactification of $\omega^* \setminus \Set{x}$.

It follows immediately that the Stone-\v{C}ech compactification of $\omega^* \setminus \Set{x}$ cannot coincide with its one-point compactification. Therefore, $\omega^* \setminus \Set{x}$ is not $C^*$-embedded in $\omega^*$.
\end{proof}

Analogues of these results for $S_\kappa$ were observed by Negrepontis \cite[14.2]{Ultrafilters}.

\begin{mythm}
\label{butterfly2}
Assume $\kappa = \kappa^{<\kappa}$. Every point in $S_\kappa$ is a strong butterfly point. \qed
\end{mythm}

\begin{mycor}
\label{nonStone2}
Assume $\kappa = \kappa^{<\kappa}$. Every card of $S_\kappa$ has a two-point compactification. In particular, no card $S_\kappa \setminus \Set{x}$ is $C^*$-embedded in $S_\kappa$. \qed
\end{mycor}

We have established that cards of $\omega^*$ under CH, and cards of $S_\kappa$ under $\kappa = \kappa^{<\kappa}$ have non-trivial finite compactifications. We now give a precise characterisation how finite compactifications of cards of these spaces look like. Our aim is to prove the following pair of theorems.

\begin{mythm}
\label{ClassificationCompactifications}
\textnormal{[CH].} For every $x \in \omega^*$ we have that
\begin{enumerate}[(a)]
\item $\omega^* \setminus \Set{x}$ has arbitrarily large finite compactifications,
\item every finite compactification of $\omega^* \setminus \Set{x}$ is homeomorphic to $\omega^*$, and
\item for every finite compactification, all but at most one point at infinity are $P$-points.
\end{enumerate}
\end{mythm}

The case for $S_\kappa$ looks exactly the same.
\begin{mythm}
\label{ClassificationCompactificationsSkappa}
Assume $\kappa = \kappa^{<\kappa}$. For every $x \in S_\kappa$ we have that
\begin{enumerate}[(a)]
\item $S_\kappa \setminus \Set{x}$ has arbitrarily large finite compactifications,
\item every finite compactification of $S_\kappa \setminus \Set{x}$ is homeomorphic to $S_\kappa$, and
\item for every finite compactification, all but at most one point at infinity are $P_\kappa$-points.
\end{enumerate}
\end{mythm}

To prove these theorems, we begin with a sufficient condition for zero-dimensional locally compact Hausdorff spaces to have only one homeomorphism type amongst their finite compactifications.

\begin{mylem}
\label{theorem1999}
Let $X$ be a zero-dimensional compact Hausdorff space such that $X \oplus X$ is homeomorphic to $X$ and
\begin{enumerate}
\item[$(\star)$]  for every point $x$ of $X$, the one-point compactification of any clopen non-compact subset of $X \setminus \Set{x}$ is homeomorphic to $X$.
\end{enumerate}
Then, for all $x$, all finite compactifications of $X \setminus \Set{x}$ are homeomorphic to $X$.
\end{mylem}

\begin{proof}
Let $Z$ be a finite compactification of $X \setminus \Set{x}$ with remainder consisting of points $\infty_1,\ldots, \infty_n$. By \cite[2.3]{Woods}, every finite compactification of a locally compact zero-dimensional space is zero-dimensional. Hence, there is a partition of $Z$ into $n$ disjoint clopen sets $A_i$ such that $\infty_i \in A_i$.

The set $A_i \setminus \Set{\infty_i}$ is a clopen non-compact subspace of $X \setminus \Set{x}$. Therefore, by property ($\star$) and uniqueness of the one-point compactification, it follows that $A_i$ is homeomorphic to $X$. This proves, after applying $X \oplus X \cong X$ iteratively, that $Z$ is homeomorphic to $X$.
\end{proof}

This lemma lies at the heart of our proofs for Theorems  \ref{ClassificationCompactifications} and \ref{ClassificationCompactificationsSkappa}. Let us see that it applies to both $\omega^*$ under CH, and to $S_\kappa$ assuming $\kappa = \kappa^{<\kappa}$.

\begin{mylem}[{\cite[3.4]{comppaper}}]
\label{lemma123}
\textnormal{[CH].} The space $\omega^*$ has property $(\star)$, i.e.\ for every $x$ the one-point compactification of a clopen non-compact subset of $\omega^* \setminus \Set{x}$ is homeomorphic to $\omega^*$.
\end{mylem}

\begin{proof}
Let $A$ be a clopen non-compact subset of $\omega^* \setminus \Set{x}$. Taking $A \cup \Set{x}$, a closed subset of $\omega^*$, as representative of its one-point compactification, we use fact $(2)$ from Section \ref{section3} to see that it is a zero-dimensional compact $F$-space of weight $\mathfrak{c}$ without isolated points.

Suppose that $U \subset A \cup \Set{x}$ is a non-empty $G_\delta$-set. If $U$ has empty intersection with $A$, then the singleton $U=\Set{x}$ is a $G_\delta$-point, and hence has countable character in the compact Hausdorff space $A \cup \Set{x}$ \cite[3.3.4]{Eng}. It follows that there is a non-trivial sequence in $\omega^*$ converging to $x$, contradicting fact $(3)$ from Section \ref{section3}. Thus, $U$ intersects the open set $A$ and their intersection is a non-empty $G_\delta$-set of $\omega^*$ with non-empty interior.

Applying Parovi\v{c}enko's Theorem \ref{parovtheorem} completes the proof.
\end{proof}

For a proof of the result in case of the space $S_\kappa$, we refer the reader to our earlier paper \cite{comppaper}.

\begin{mylem}[{\cite[4.4]{comppaper}}]
\label{lemma123b}
Assume $\kappa = \kappa^{<\kappa}$. The space $S_\kappa$ has property $(\star)$, i.e.\ for every $x$ the one-point compactification of a clopen non-compact subset of $S_\kappa \setminus \Set{x}$ is homeomorphic to $S_\kappa$. \qed
\end{mylem}

Thus, we have verified that Lemma \ref{theorem1999} does apply to our spaces $\omega^*$ and $S_\kappa$. Surprisingly, despite its strong assumptions, the lemma itself applies to a variety of interesting spaces.

Spaces which only have $\lambda$ different homeomorphism types amongst their open subspaces (for some cardinal $\lambda$) are said to be of \emph{diversity} $\lambda$ \cite{diversity}. One checks that Lemma \ref{theorem1999} applies to all compact Hausdorff spaces of diversity two, which are known to be zero-dimensional \cite{zerodiv}. In particular, Lemma \ref{theorem1999} applies to the Cantor space $C$, which can be characterised as the unique compact metrizable space of diversity two \cite{Cantor}. It also applies to the Alexandroff Double Arrow space $D$ and to the product $D \times C$. Incidentally, these examples are also non-reconstructible \cite[2.5]{recpaper}.

In a compact Hausdorff space $X$ of diversity two, any subspace $X \setminus \Set{x}$ is homeomorphic to $X \setminus \Set{x_1,\ldots,x_n}$ and therefore has arbitrarily large finite compactifications. This is when Lemma \ref{theorem1999} is most valuable. Our next lemma shows that not much is needed for this scenario to occur. The proof is a simple induction.

\begin{mylem}
\label{lemm2}
Let $X$ be a topological space such that for all $x$, all finite compactifications of $X \setminus \Set{x}$ are homeomorphic to $X$. If all spaces $X \setminus \Set{x}$ have two-point compactifications, they have arbitrarily large finite compactifications. \qed
\end{mylem}

The following example of the Cantor cube $2^\kappa$ for uncountable $\kappa$ shows that the assumptions in Lemma \ref{lemm2} cannot be considerably weakened. Since $\beta (2^\kappa \setminus \Set{x})=2^\kappa$ \cite[Thm.\ 2]{Glicksberg}, these spaces have a unique compactification. The cube $2^\kappa$ is a zero-dimensional compact Hausdorff space with $2^\kappa \cong 2^\kappa  \oplus 2^\kappa$. For property $(\star)$, let $A \subset 2^\kappa \setminus \Set{x}$ be a clopen non-compact subset. Since $2^\kappa \setminus \Set{x}$ does not have a 2-point compactification, $A \cup \Set{x}$ must be clopen in $2^\kappa$. But every clopen set of $2^\kappa$ can be written as a disjoint union of finitely many product-basic open sets, which are homeomorphic to $2^\kappa$. Hence $A \cup \Set{x} \cong 2^\kappa$. We conclude that Lemma \ref{theorem1999} applies, but restricts to the obvious assertion that the one-point compactification of $2^\kappa \setminus \Set{x}$ is homeomorphic to $2^\kappa$.

Now finally, with these lemmas established, it is not difficult to give proofs of the main theorems in this section.

\begin{proof}[Proof of Theorem \ref{ClassificationCompactifications}]
Assertion $(b)$ is an immediate consequence of Lemmas \ref{theorem1999} and \ref{lemma123}. For $(a)$, note that Corollary \ref{nonStone} implies that every card of $\omega^*$ has a two-point compactification, which is homeomorphic to $\omega^*$ by $(b)$. Therefore, $(a)$ now follows from Lemma \ref{lemm2}.

For assertion $(c)$, suppose there is a finite compactification $Z$ of $\omega^* \setminus \Set{x}$ containing two non-$P$-points $\infty_1$ and $\infty_2$ at infinity. Then there are disjoint open $F_\sigma$-sets $F_1$ and $F_2$ in $Z$ with $F_i \subset \omega^* \setminus \Set{x}$ containing $\infty_1$ and $\infty_2$ in their respective boundaries. However, in $\omega^*$ the disjoint non-compact open $F_\sigma$-sets $F_1$ and $F_2$ both limit onto $x$. This contradicts the $F$-space property of $\omega^*$.
\end{proof}

In ZFC, the above argument still shows that every finite compactification of $\omega^* \setminus \Set{x}$ is a Parovi\v{c}enko space of weight $\mathfrak{c}$ such that at most one point at infinity is not a $P$-point. However, one cannot decide in ZFC alone whether there are finite compactifications of $\omega^* \setminus \Set{x}$ other than the one-point compactification \cite{douwenkunenmill}.

\begin{proof}[Proof of Theorem \ref{ClassificationCompactificationsSkappa}]
Assertions $(a)$ and $(b)$ follow as in the previous proof.

The proof of $(c)$ uses the same idea as in the case of $\omega^*$. Suppose there is a finite compactification $Z$ of $S_\kappa \setminus \Set{x}$ containing two non-$P_\kappa$-points $\infty_1$ and $\infty_2$ at infinity. Then there are disjoint open subsets $F_1$ and $F_2$ in $Z$ of type less than $\kappa$ with $F_i \subset S_\kappa \setminus \Set{x}$ that contain $\infty_1$ and $\infty_2$ in their respective boundaries. However, in $S_\kappa$ the disjoint non-compact open sets $F_1$ and $F_2$ of type less than $\kappa$ both  limit onto $x$, contradicting the $F_\kappa$-space property.
\end{proof}

\section{\texorpdfstring{Non-normality of cards of $\omega^*$ and $S_\kappa$}{Non-normality of cards}}
\label{sectionnormality}

This section contains proofs that under CH, every card of $\omega^*$ is non-normal, and that for uncountable $\kappa$ with $\kappa = \kappa^{<\kappa}$, every card of $S_\kappa$ is non-normal.

In case of $\omega^*$, this result is originally due to Rajagopalan and Warren. Proofs can be found in \cite{easy, raj,Warren}. An account of \cite{raj} is contained in \cite[7.2-7.4]{Walker}. In the first part of this section, we advertise a different, very elegant proof of this classical theorem. The proof builds on ideas from \cite{logunov, sapirowskii, terasawa}, and is mentioned, without details, in \cite[Rmk.\ 3]{terasawa}. The result that also cards of $S_\kappa$ are non-normal is new. Our proof generalises the approach of \cite{Warren} for $\omega^*$.

To begin, we note that normality of cards of the spaces $\beta \omega$ and $\omega^*$ are in fact equivalent problems, in the sense that when deleting a point of the remainder of $\beta \omega$, the card of $\beta \omega$ is normal if and only if the corresponding card of $\omega^*$ is normal. This result is folklore, but as no proof is available in the standard literature we give one in the next lemma.

\begin{mylem}
\label{equivalentnormality}
Let $x \in \omega^*$. Then $\beta \omega \setminus \Set{x}$ is normal if and only if $\omega^* \setminus \Set{x}$ is normal.
\end{mylem}

\begin{proof}
If $\beta \omega \setminus \Set{x}$ is a normal space then its closed subspace $\omega^* \setminus \Set{x}$ inherits normality.

For the converse implication, assume that $\omega^* \setminus \Set{x}$ is normal and let $C_1$ and $C_2$ be disjoint closed subsets of $\beta \omega \setminus \Set{x}$. Let $C'_i = C_i \cap \omega^*$ be the part of each closed set lying in the remainder of $\omega^*$. Since $\omega^* \setminus \Set{x}$ is normal, there are open subsets $W_1$ and $W_2$ of $\omega^* \setminus \Set{x}$ with disjoint closures such that $C'_i \subset W_i$. Note that all open sets $V_1$ and $V_2$ in $\beta \omega \setminus \Set{x}$ with $W_i = V_i \cap \omega^*$ can only intersect in a finite subset of $\omega \subset \beta \omega$. Hence $U_1=V_1 \setminus V_2$ and $U_2=V_2 \setminus V_1$ are disjoint open subsets of $\beta \omega \setminus \Set{x}$ containing $C'_1$ and $C'_2$ respectively.

But now, as any subset of $\omega$ is open in $\beta \omega$, the sets
$$ (U_1 \setminus C_2) \cup (C_1\cap \omega) \; \text{ and } \; (U_2 \setminus C_1) \cup (C_2\cap \omega)$$
are disjoint open subsets of $\beta \omega \setminus \Set{x}$ containing $C_1$ and $C_2$ respectively. This completes the proof that $\beta \omega \setminus \Set{x}$ is normal.
\end{proof}

\begin{mythm}
\label{nonnormalcards}
\textnormal{[CH].} Every card $\omega^* \setminus \Set{x}$ of $\omega^*$ is non-normal.
\end{mythm}

\begin{proof}
Let $x \in \omega^*$. By Lemma \ref{equivalentnormality} it suffices to show that $\beta \omega \setminus \Set{x}$ is non-normal.
Assume for a contradiction that $\beta \omega \setminus \Set{x}$ is normal. By Tietze's Theorem \cite[2.1.8]{Eng}, the closed subset $\omega^* \setminus \Set{x}$ is $C^*$-embedded in $\beta \omega \setminus \Set{x}$, which in turn is $C^*$-embedded in $\beta \omega$ \cite[3.6.9]{Eng}.

Thus, $\omega^* \setminus \Set{x}$ is $C^*$-embedded in $\omega^*$, contradicting Corollary \ref{nonStone}.
\end{proof}

We now come to the result that cards of $S_\kappa$ are non-normal. We will see that Lemma \ref{lemma123b} implies that without loss of generality we may focus our attention on cards obtained by deleting a $P_\kappa$-point of $S_\kappa$. From there on, we adapt Warren's result \cite[I.1]{Warren} that under CH, cards of $\omega^*$ obtained by deleting $P$-points are non-normal.

\begin{mythm}
\label{nonnormalskappa}
Assume $\kappa = \kappa^{<\kappa}$. If $\kappa$ is uncountable then every card $S_\kappa \setminus \Set{x}$ of $S_\kappa$ is non-normal.
\end{mythm}

\begin{proof}
It suffices to prove the theorem for cards that have been obtained by deleting a $P_\kappa$-point of $S_\kappa$. To see this, we use that by Theorem \ref{butterfly2}, every point $x \in S_\kappa$ is a butterfly point of $S_\kappa$ with wings $A$ and $B$. Using Theorem \ref{ClassificationCompactificationsSkappa}(c), we may assume that $x$ is a $P_\kappa$-point with respect to $A \cup \Set{x}$, which in turn is homeomorphic to $S_\kappa$ by Lemma \ref{lemma123b}. Thus, if $S_\kappa$ removed a $P_\kappa$-point is non-normal, then $S_\kappa \setminus \Set{x}$ contains the closed, non-normal subspace $A$, and hence is itself non-normal.

So let $p$ be a $P_\kappa$-point in $S_\kappa$. We show that $S_\kappa \setminus \Set{p}$ is non-normal. Fix a strictly decreasing neighbourhood base $\set{U_\alpha}:{\alpha < \kappa}$ of $p$ consisting of clopen sets. Pick $P_\kappa$-points $p_\alpha$ inside the non-empty sets $V_\alpha = U_\alpha \setminus U_{\alpha + 1}$, which is possible as $V_\alpha$ is homeomorphic to $S_\kappa$. Again, for each $p_\alpha \in V_\alpha$ we fix a nested neighbourhood base $\set{V_{\alpha,\beta}}:{\beta < \kappa}$ of clopen sets, such that $V_{\alpha,0}=V_\alpha$.

We now describe two closed disjoint sets $A$ and $B$ of $S_\kappa \setminus \Set{p}$ that cannot be separated by open sets, showing that this space is non-normal. Define, for each limit ordinal $\lambda<\kappa$, the sets
$$B_\lambda = \closure{\bigcup_{\alpha < \lambda} (V_\alpha \setminus V_{\alpha,\lambda})} \cap \bigcap_{\alpha < \lambda}U_\alpha.$$
The closures are of course taken in $S_\kappa \setminus \Set{p}$.
Then put
$$A=\closure{\set{p_\alpha}:{\alpha < \kappa}} \quad \text{and} \quad B = \bigcup_{\lambda < \kappa} B_\lambda.$$

Let us see why $A$ and $B$ are disjoint. Consider the disjoint sets $\bigcup_{\alpha < \lambda} V_{\alpha,\lambda}$ and $\bigcup_{\alpha < \lambda} V_\alpha \setminus V_{\alpha,\lambda}$, the first of which being a superset of $\set{p_\alpha}:{\alpha < \lambda}$. Both sets are of $S_\kappa$-type less than $\kappa$ and hence have disjoint closures in $S_\kappa$ by fact $(1')$ from Section \ref{section3}. It follows that $A$ and $B$ are disjoint, since if $q \in A \cap B$ then $q \notin U_\lambda$ for some limit ordinal $\lambda$, and we obtain a contradiction from
$$q \in A \cap B \cap (S_\kappa \setminus U_\lambda) \subset \closure{\bigcup_{\alpha < \lambda} V_{\alpha,\lambda}}  \cap \closure{ \bigcup_{\alpha < \lambda} V_\alpha \setminus V_{\alpha,\lambda}} = \emptyset.$$

We now show that $B$ is closed. Suppose that $q$ lies in $\closure{B}$. Since $V_\alpha \cap B = \emptyset$ for all $\alpha < \kappa$ it follows that $q \in \bigcap_{\alpha < \mu} U_\alpha \setminus U_\mu$ for some limit ordinal $\mu < \kappa$. Note that $$B=\bigcup_{\lambda < \mu} B_\lambda \cup B_\mu \cup \bigcup_{\mu < \lambda < \kappa} B_\lambda.$$
Since the last factor is a subset of the closed set $U_\mu$, the point $q$ is contained in the closure of the first two factors. However, it follows from the construction that $\closure{\bigcup_{\lambda < \mu} B_\lambda} \cap \bigcap_{\alpha < \mu} U_\alpha \setminus U_\mu \subset B_\mu$. So $q$ lies in $\closure{B_\mu}=  B_\mu$. Hence $q \in B$, and we have shown that $B$ is closed.

To complete the proof it remains to show that $A$ and $B$ cannot be separated by open sets. So let $U$ and $V$ be open sets of $S_\kappa \setminus \Set{p}$ containing $A$ and $B$ respectively. For every ordinal $\alpha<\kappa$ there exists $\beta_\alpha > \alpha$ such that $V_{\alpha,{\beta_\alpha}} \subset  U$. Since $\kappa=\kappa^{<\kappa}$ implies that $\kappa$ is regular \cite[1.27]{Ultrafilters}, the increasing sequence defined by $\alpha_0=0$ and $\alpha_n=\beta_{\alpha_{n-1}}$ has a supremum $\gamma < \kappa$. Consider the set
$$W = \bigcup_{n \in \omega} (V_{\alpha_n,\beta_{\alpha_n}} \setminus V_{\alpha_n,\gamma}).$$
It follows from our construction that
$$W \subset U \quad \text{and} \quad W \subset \bigcup_{n \in \omega} (V_{\alpha_n} \setminus V_{\alpha_n,\gamma}).$$
Let us see that $\closure{W} \cap \bigcap_{\alpha < \gamma}U_\alpha$ is a non-empty subset of $B_\gamma$. For this, we only have to show that $\closure{W}$ intersects $\bigcap_{\alpha < \gamma}U_\alpha =\bigcap_{n \in \omega}U_{\alpha_n}$. But this holds, since otherwise the collection $\set{ S_\kappa \setminus U_{\alpha_n}}:{n \in \omega}$ forms an open cover of the compact set $\closure{W}$, yielding a contradiction.

It follows that $V$ is a neighbourhood of every point in $\closure{W} \cap \bigcap_{\alpha < \gamma}U_\alpha$, and therefore that $V\cap W \neq \emptyset$. Since
 $V \cap U \supseteq V \cap W$ we see that $U$ and $V$ cannot be disjoint, completing the proof.
\end{proof}

\section{\texorpdfstring{Reconstruction results for $\omega^*$ and $S_\kappa$}{Reconstruction of w* and Skappa}}
\label{section4}

This section contains our reconstruction results for the spaces $\omega^*$ and $S_\kappa$. We will see that it is independent of ZFC whether the space $\omega^*$ is reconstructible. For example, $\omega^*$ is reconstructible in models where $\omega^*$ is the Stone-\v{C}ech compactification of one of its cards, and is not reconstructible in models where the Continuum Hypothesis holds.

Generalising the behaviour of $\omega^*$ under CH, we show further below that assuming $\kappa = \kappa^{<\kappa}$, the spaces $S_\kappa$ are always non-reconstructible.

\begin{mythm}
\label{mainresultsrec}
\textnormal{[CH].} The space $\omega^*$ is non-reconstructible. For a $P$-point $p$, the space $\omega^* \setminus \Set{p}$ is a non-homeomorphic reconstruction of $\omega^*$.
\end{mythm}

For the proof we need three lemmas describing the behaviour of quotients of Parovi\v{c}enko spaces $X$ when identifying a subset $A$ with a single point. Write $X/A$ for the quotient space induced by the partition $\Set{A} \cup \set{\Set{x}}:{x \in X \setminus A}$.

\begin{mylem}
\label{quotient1}
Let $X$ be a compact Hausdorff space and $A \subset X$ a closed, non-open subset of $X$. Then $X / A$ is a one-point compactification of $X \setminus A$. Moreover, if $X$ is zero-dimensional, then so is $X/A$.
\end{mylem}
\begin{proof}
First, a quotient space of a compact space is compact. Further, $X/A$ is Hausdorff as $X$ is regular and $A$ is closed.
Since $A$ is not open, the map $X \setminus A \hookrightarrow X/A$, sending $x \mapsto \Set{x}$ is a dense embedding with a one-point remainder.

For zero-dimensionality, we show that $A \subset X$ has a neighbourhood base of clopen sets. So let $U$ be an open neighbourhood of $A$. Assuming that $X$ is zero-dimensional, for every $x \in A$ there is a clopen set such that  $x \in C(x) \subseteq U$. The clopen cover $\set{C(x)}:{x \in A}$ of the compact set $A$ has a finite subcover. Its union is a clopen set between $A$ and $U$.
\end{proof}

The next lemma shows in which cases quotients preserve the $G_\delta$-property of Parovi\v{c}enko spaces. A similar lemma appears without proof in \cite[1.4.2]{Intro}. Note that if $X$ is compact Hausdorff and $A \subset X$ is closed, the quotient map $\pi \colon X \to X/A$ is a continuous map from a compact space to a Hausdorff space, and therefore closed.

\begin{mylem}
\label{quotient2}
Suppose $X$ has the property that non-empty $G_\delta$-sets have non-empty interior. Let $A \subset X$ be a closed, nowhere dense subset of $X$. Then $X / A$ also has the property that non-empty $G_\delta$-sets have non-empty interior.
\end{mylem}
\begin{proof}
Let $U$ be a non-empty $G_\delta$ of $X/A$. We have to show that it has non-empty interior. Since $\pi$ is continuous and surjective, $\pi^{-1}(U)$ is a non-empty $G_\delta$-set of $X$. By assumption, it has non-empty interior. Since $A$ is closed and nowhere dense, the set $\pi^{-1}(U) \setminus A$ also has non-empty interior. Observing that $\pi(\interior{\pi^{-1}(U) \setminus A})$ is an open subset of $U$ completes the proof.
\end{proof}

Our last lemma tells us under which conditions collapsing a subset to a single point leaves the $F$-space property intact. The result is a slight generalisation of \cite[1.4.1]{Intro}.

\begin{mylem}
\label{quotient3}
Let $X$ be a compact $F$-space and $A\subset X$ a closed subset containing at most one non-$P$-point of $X$. Then $X/A$ is an $F$-space.
\end{mylem}

\begin{proof}
By Lemma \ref{quotient1}, the space $X/A$ is normal. To establish the $F$-space property, it therefore suffices, using fact $(1)$ from Section \ref{section3}, to verify that disjoint open $F_\sigma$-sets have disjoint closures.

So let $U$ and $V$ be disjoint open $F_\sigma$-sets of $X/A$. Since $\pi^{-1}(U)$ and $\pi^{-1}(V)$ are disjoint open $F_\sigma$-sets of $X$, they have disjoint closures in $X$.

Suppose that $\Set{A} \in U \cup V$. Without loss of generality, we have $A \subset \pi^{-1}(U)$ and hence $\pi(\closure{\pi^{-1}(U)}) \cap \pi(\closure{\pi^{-1}(V)}) = \emptyset$. Since $\pi$ is a closed surjective map, we have $\closure{U} \subseteq \pi(\closure{\pi^{-1}(U)})$ and hence $\closure{U} \cap \closure{V} = \emptyset$.

Now suppose that $\Set{A} \notin U \cup V$. Then $A$ does not intersect $\pi^{-1}(U) \cup \pi^{-1}(V)$. Note that for all $P$-points $p \in A$ it follows from fact $(4)$ in Section \ref{section3} that
$$p \notin \closure{\pi^{-1}(U)} \cup \closure{\pi^{-1}(V)}.$$
 Finally, since $\pi^{-1}(U)$ and $\pi^{-1}(V)$ have disjoint closures in $X$, the single non-$P$-point of $A$ cannot be contained in both of them. Thus, we may assume without loss of generality that $\Set{A} \notin \pi(\closure{\pi^{-1}(U)})$. Therefore, $\pi(\closure{\pi^{-1}(U)}) \cap \pi(\closure{\pi^{-1}(V)}) = \emptyset$, implying, as before, that $U$ and $V$ have disjoint closures in $X/A$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{mainresultsrec}]
We prove that for a $P$-point $p$ of $\omega^*$, the space $\omega^* \setminus \Set{p}$ is a non-homeomorphic reconstruction of $\omega^*$. Let us first show $\singletonDeletion{\omega^* \setminus \Set{p}} \subseteq \singletonDeletion{\omega^*}$.

For this inclusion, pick any card $\omega^* \setminus \Set{p,x}$ in $\singletonDeletion{\omega^* \setminus \Set{p}}$. We claim that its one-point compactification $X=\Set{\infty} \cup \parentheses{\omega^*\setminus \Set{p,x}}$ is a Parovi\v{c}enko space. Then, by Theorem \ref{parovtheorem}, there is a homeomorphism $f \colon X \to \omega^*$. It follows
$$\omega^* \setminus \Set{p,x} \cong X \setminus \Set{\infty} \cong \omega^* \setminus \Set{f(\infty)},$$
establishing that $\omega^* \setminus \Set{p,x} \in \singletonDeletion{\omega^*}$.

To see that $X$ is a Parovi\v{c}enko space note that $X$ is a compact space of weight $\mathfrak{c}$ without isolated points. By Lemma \ref{quotient1}, we may take $\omega^*/A$ with $A = \Set{p,x}$ as a representative for $X$, showing that $X$ is zero-dimensional. Further, by Lemmas \ref{quotient2} and \ref{quotient3}, the space $\omega^*/A$ is an $F$-space with the property that non-empty $G_\delta$-sets have non-empty interior. Thus, $X$ is Parovi\v{c}enko, completing the proof of the first inclusion.

We now establish the reverse inclusion $\singletonDeletion{\omega^* \setminus \Set{p}} \supseteq \singletonDeletion{\omega^*}$. For this, let $\omega^* \setminus \Set{x}$ be any card in $\singletonDeletion{\omega^*}$. It follows from Theorem \ref{ClassificationCompactifications} that there exist points $\infty_1$ and $\infty_2$ of $\omega^*$, of which $\infty_1$ is a $P$-point, such that
$$\omega^* \setminus \Set{x} \cong \omega^* \setminus \Set{\infty_1,\infty_2}.$$
By fact $(5)$ from Section \ref{section3}, there exists a homeomorphism $f$ of $\omega^*$ carrying $\infty_1$ to $p$. Then $\omega^* \setminus \Set{x} \cong \omega^* \setminus \Set{p,f(\infty_2)}$, and hence $\omega^* \setminus \Set{x}$ is a card in $\singletonDeletion{\omega^* \setminus \Set{p}}$.
\end{proof}

The proof that $S_\kappa$ is non-reconstructible is very similar. The following two lemmas are straightforward adaptions of Lemma \ref{quotient2} and \ref{quotient3} respectively.

\begin{mylem}
\label{quotient4}
Suppose $X$ has the property that every non-empty intersection of fewer than $\kappa$ many open sets has non-empty interior. Let $A \subset X$ be a closed, nowhere dense subset of $X$. Then $X / A$ also has the property that every non-empty intersection of fewer than $\kappa$ many open sets has non-empty interior. \qed
\end{mylem}

\begin{mylem}
\label{quotient5}
Let $X$ be a compact $F_\kappa$-space and $A\subset X$ a closed subset containing at most one non-$P_\kappa$-point of $X$. Then $X/A$ is an $F_\kappa$-space. \qed
\end{mylem}

\begin{mythm}
\label{mainresultsrec2}
Assume $\kappa = \kappa^{<\kappa}$. The space $S_\kappa$ is non-reconstructible. Indeed, for a $P_\kappa$-point $p$, the space $S_\kappa \setminus \Set{p}$ is a non-homeomorphic reconstruction of $S_\kappa$.
\end{mythm}

\begin{proof}
To prove the inclusion $\singletonDeletion{S_\kappa \setminus \Set{p}} \subseteq \singletonDeletion{S_\kappa}$, pick any card $S_\kappa \setminus \Set{p,x}$ in $\singletonDeletion{S_\kappa \setminus \Set{p}}$. Using Lemmas \ref{quotient4} and \ref{quotient5} we see that its one-point compactification $X=\Set{\infty} \cup \parentheses{S_\kappa\setminus \Set{p,x}}$ is a $\kappa$-Parovi\v{c}enko space. By Theorem \ref{negre} there is a homeomorphism $f \colon X \to S_\kappa$. It follows
$$S_\kappa \setminus \Set{p,x} \cong X \setminus \Set{\infty} \cong S_\kappa \setminus \Set{f(\infty)},$$
establishing that $S_\kappa \setminus \Set{p,x} \in \singletonDeletion{S_\kappa}$.

For the reverse inclusion, let $S_\kappa \setminus \Set{x}$ be a card in $\singletonDeletion{S_\kappa}$. It follows from Theorem \ref{ClassificationCompactificationsSkappa} that there are points $\infty_1$ and $\infty_2$ of $S_\kappa$, with $\infty_1$ being a $P_\kappa$-point, such that
$$S_\kappa \setminus \Set{x} \cong S_\kappa \setminus \Set{\infty_1,\infty_2}.$$
By fact $(5')$ from Section \ref{section3}, there exists a homeomorphism $f$ of $S_\kappa$ carrying $\infty_1$ to $p$. Then $S_\kappa \setminus \Set{x} \cong S_\kappa \setminus \Set{p,f(\infty_2)}$, and hence $S_\kappa \setminus \Set{x}$ is a card in $\singletonDeletion{S_\kappa \setminus \Set{p}}$.
\end{proof}

We conclude this section with the result that $\omega^*$ is consistently reconstructible. In particular, together with the results above we see that the question whether $\omega^*$ is reconstructible is independent of the axioms of set theory ZFC. Note also that for showing that $\omega^*$ is non-reconstructible, the assumption CH cannot be weakened to Martin's axiom (MA).

\begin{mythm}[{\cite{douwenkunenmill}}]
\label{douwenkunenmilltheorem}
It is consistent with $MA+\mathfrak{c}=\omega_2$ that for all $x\in \omega^*$ we have $\beta(\omega^* \setminus \Set{x})=\omega^*$. \qed
\end{mythm}
\begin{mythm}
It is consistent with $MA+\mathfrak{c}=\omega_2$ that $\omega^*$ is reconstructible.
\end{mythm}
\begin{proof}
It is shown in \cite[5.4]{recpaper} that every compact Hausdorff space arising as a non-trivial Stone-\v{C}ech compactification is reconstructible. Hence, the reconstruction result follows from the previous theorem.
\end{proof}

\section{Normality is consistently non-reconstructible}
\label{sectionalltogether}

Theorem \ref{mainresultsrec} established that under CH, the spaces $\omega^*$ and $\omega^* \setminus \Set{p}$ for a $P$-point $p$ are non-homeomorphic reconstructions of each other. Since under CH, the space $\omega^* \setminus \Set{p}$ is non-normal by Theorem \ref{nonnormalcards}, this gives the desired result that normality is consistently non-reconstructible. Also, in presence of Hausdorffness, compactness implies paracompactness, which in turn implies normality \cite[5.1.1 \& 5.1.18]{Eng}. Hence, it is also consistent that paracompactness is non-reconstructible.

More generally, we have the following theorem.

\begin{mythm}
\label{thedefinitenormalityresult}
The existence of an uncountable cardinal $\kappa$ with the property $\kappa=\kappa^{<\kappa}$ implies that normality and paracompactness are not reconstructible.
\end{mythm}

\begin{proof}
The space $S_\kappa$ is compact Hausdorff, hence paracompact and normal. By Theorem \ref{nonnormalskappa}, the space $S_\kappa \setminus \Set{p}$ for a $P_\kappa$-point $p$ is non-normal and non-paracompact. However, by Theorem \ref{mainresultsrec2}, both spaces are reconstructions of each other. Therefore, the properties of being normal or paracompact are not reconstructible.
 \end{proof}

\section{Questions}
\label{sectionquestions}

In the previous section we have shown that normality and paracompactness are consistently non-reconstructible. Are these results true in ZFC?

\begin{myquest}
Are normality or paracompactness non-reconstructible properties?
\end{myquest}

Next, we have shown that it is consistent with the negation of CH that $\omega^*$ is reconstructible. Is this always the case? Note that our present proof of the fact that $\omega^*$ is non-reconstructible under CH uses, on several occasions, the full power of Parovi\v{c}enko's theorem---which itself is equivalent to CH.

\begin{myquest}
Is CH equivalent to the assertion that $\omega^*$ is non-reconstructible?
\end{myquest}

Our last question asks whether in Theorem \ref{mainresultsrec}, we can tell which point precisely one has to delete from $\omega^* \setminus \Set{p}$ in order to obtain a given card of $\omega^*$. For example, it is easy to see that under CH, if $q$ is a further $P$-point then $\omega^* \setminus \Set{p,q} \cong \omega^* \setminus \Set{q}$. Does this behaviour occur for all points of $\omega^*$?

\begin{myquest}
\textnormal{[CH].} Let $p$ be a $P$-point of $\omega^*$. Is it true that for all $x$ we have $\omega^* \setminus \Set{p,x} \cong \omega^* \setminus \Set{x}$?
\end{myquest}

redpajama_set_name: RedPajamaArXiv


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A graduate of Hartford Public High, Remigino returned to his alma mater as a physical education teacher in 1953 after winning gold medals in the 100-meter dash and the 4x100-meter relay at the 1952 Olympic Games in Helsinki, Finland. Remigino has coached the boys and girls cross country, indoor and outdoor track teams, winning over 85 per cent of his dual meets. His track teams have won a total of 31 state titles - 21 outdoor and 10 indoor - and 25 city championships, including the last 18 in a row.
After leading Bakersfield High to four top-10 finishes in the single-class California state tournament over the previous seven seasons, David East watched his Drillers rewrite the California record books this year.
A month ago, Bakersfield scored 226.5 points in the state tournament, winning the state title by 125 points. Nine Bakersfield wrestlers finished in the top six, seven advanced to the finals, and four were crowned champions. All are California state records. Bakersfield was ranked No. 5 in the nation this season. In addition, one of East's four state champions, 125-pounder Darrell Vasquez, became the first four-time state champion in the history of California high school wrestling.
East is retiring from coaching this year after 27 years, 24 as a head coach. A graduate of Bakersfield High and Cal-State Bakersfield, where he was a member of the school's first Division II national championship team, he was head coach for five years at Maricopa High and four years at West Bakersfield High before coming to Bakersfield High. After three years as an assistant to Steve Varner, East became head coach in 1988 and led the Drillers to 14 consecutive league titles. In addition to this year's state title, his teams finished second in the state in 2001 and third in 1995. He coached at least one state placewinner in each of his final nine seasons, and overall he coached 31 state placewinners and seven state champions. His teams were undefeated in dual meets his final two seasons, and his career record was 204-12.
Jim Jackson is only the second head coach in the 26-year history of Apple Valley High's fabulously successful wrestling program. Jackson has been part of the Eagles' coaching staff for 22 of those 26 years. After 10 years as an assistant, he became co-head coach of the Eagles in 1991, serving alongside his mentor, Bill Demaray, who started the program when the school opened in 1976 and coached it to six big-class Minnesota dual state titles. Jackson was part of three of those teams as a co-head coach, in 1991, 1994, and 1995. When Demaray stepped down after the 1995 season after 20 years at the helm, Jackson became the head coach and has maintained the Eagles' position at the top of Minnesota high school wrestling.
Jackson has coached Apple Valley to the past four Class AAA dual state titles and five state titles overall in his seven years as head coach, with his other two teams finishing second and third in the state. This year's team finished 39-0 in dual meets, going out of state to record dual meet victories over national powerhouses Iowa City and Lewis Central of Iowa, Broken Arrow and El Reno of Oklahoma, and Platte County of Missouri, and was ranked among the nation's top three teams for the second time in three years. His overall dual-meet record is 226-9-3, including a current 60-match winning streak over the past two seasons.
Jackson's wrestlers also have excelled in the individual state tournament. In seven years, he has coached 46 state placewinners and 16 state champions. He also was selected National Coach of the Year by Wrestling USA magazine in 2001.
Houle coaches five sports at Mountain View High - girls basketball, boys and girls cross country, and boys and girls track and field - and has had unparalleled success in all five. His girls basketball team won its ninth state title this year, finishing No. 13 in the USA Today Super 25 (last year's team finished No. 2). In February, he won his 1,000th contest in the five sports and, starting in 1985, has won at least one state championship in every school year since. His 55 overall state titles (nine in girls basketball, 11 in boys cross country, 14 in girls cross country, eight in boys track and field, and 13 in girls track and field) are a national record for one coach. In the 1997-98 school year, all five of Houle's teams won state titles, another national record, and only a runner-up finish by his boys track and field team prevented him from repeating that feat this year.
Arbogast has coached Bingham's boys and girls cross country teams since coming to the school as an English teacher in 1980, and has coached the boys and girls track and field teams as well since 1982. All four teams have won Class 5A state titles under his tenure - a total of nine in all. His girls cross country teams won three consecutive titles, in 1998, 1999, and 2000, and finished second this year. His boys cross country teams also have three state titles, and were recognized as the nation's top team in 1995. His girls track and field team won its second state title this year. All four teams are unbeaten in conference meets for more than 10 consecutive years. One of the nation's top distance running coaches, Arbogast has seen more than 50 of his athletes earn college athletic scholarships during his coaching career.
Walker has been the head coach at Sanford High for 19 seasons. She led the team to back-to-back state titles in Maine's big-school Class A in 1999 and 2000, the first school in nearly a decade to win consecutive Class A titles. This past season, Sanford reached the state title game for the third consecutive year, and built a 50-match winning streak over the three seasons before seeing it come to an end in this year's state final.В Sports IllustratedВ magazine also recognized Walker's distinguished career with a mention in its "Faces in the Crowd" section this year.
Stendahl is retiring from coaching this year after a distinguished career that saw him coach both the boys and girls teams at Middleton High. In his final season, Stendahl's girls team qualified for the Division I state tournament for the fifth consecutive year and finished second, its best finish ever. His 1999 team finished third. All five of his top girls golfers were selected to the Big Eight Conference First or Second Teams. Four of Stendahl's boys teams won Division I state titles, including a three-peat in 1997, 1998, and 1999 after he won his first title in 1993. His 1991 team finished second, and his final squad this spring finished eighth.
A 1970 graduate of the University of Wisconsin-LaCrosse, Stendahl also was an outstanding kicker on the school's football team in addition to his golf success.
Under McVey's direction, St. Joseph's Academy had its best girls soccer season ever, finishing the season with a 28-0-1 record to become the first Missouri team to go through a season unbeaten. St. Joseph's Academy earned a unanimous No. 1 ranking nationally in the National Soccer Coaches Association of America (NSCAA) Girls Spring Top 20 and also by StudentВ Sports.com. In a state level, St. Joseph's Academy won its second consecutive Class 4A state title under McVey, and its fourth in six years.
In 26 years of coaching, Adams has won nearly 1,000 games while coaching softball and boys basketball. Since 1976, Adams' softball teams have a record of 629-63, and have reached the Class AA state finals every year since 1980. A national record 17 of those teams have won titles, including a streak of 11 straight from 1987-97. The National High School Athletic Coaches Association selected him as the National Coach of the Year in 1991, and he has been named conference or region Coach of the Year 15 times.
From 1976 to 1992, Adams also was the school's boys basketball coach, leading that squad to two state titles and one runner-up finish. The South Carolina Athletic Coaches Association (SCACA) named him its basketball Coach of the Year three times, and he was conference Coach of the Year seven times. The SCACA also presented Adams with its Outstanding Service Award in 1994 and 1999. He is one of three coaches to be inducted into the National Federation of State High School Associations Hall of Fame this year.
Under Beerman's direction, Ursuline Academy has been the top girls swimming program in the swimming hotbed of southwestern Ohio. He has led the Lions to 10 finishes in the state's top in the past 12 years. This year's team won its third consecutive Division I state title, its fifth overall under Beerman, who also led Ursuline Academy to state titles in 1991 and 1998. His teams also were state runners-up in 1992, 1994, 1995, 1996, and 1999.
Stevens has been head tennis coach at Beverly Hills High for 28 years, and her last two teams may have been her best. In 2000, Beverly Hills captured the CIF Division II state title for the first time in 28 years, and won the prestigious National High School All-America Tournament, held in Newport Harbor.SchoolSports.comВ named the entire Beverly Hills team as its Girls Tennis Player of the Year that season, the only time the service has ever done so. Beverly Hills High repeated as CIF Southern Section champions this year, this time in Division III.
Knudson just completed his third season as Northrop High's girls track and field coach, and presided over the three most successful seasons in school history as Northrop High won its third consecutive state team title. His team won this year's title with depth, winning just two events but picking up 10 top-five finishes overall. Northrop High now has won five state championships overall in girls track and field - the first two coming in 1981 and 1991 - and became just the second school ever to win three consecutive state titles. With a young team this year, Knudson stands a solid chance to become the first coach to win four in a row next season.
Barker led Amarillo High to a 38-2 record and the Class 5A state title this year, 16-14, 15-9 over Houston Stratford High. She has led Amarillo High to four state titles, the most ever by any Class 5A school in a state rich in volleyball tradition. Under Barker's direction, the school also won titles in 1988, 1994, and 1998, while adding a runner-up finish in 1997. In 15 years as head coach at Amarillo High, Barker's teams have a record of 486-70. She also was selected as National High School Volleyball Coach of the Year byStudentSports.com.


Another collaboration with the talented Freddy Arenas.
As a directing duo we have developed the concept and look for a unique piece around the idea of how people celebrate their holidays around the world. We chose light as the thread that take us through the story and as an element that all the cultures use to draw energy and color from celebration to celebration.


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USA Today Metro Atlanta Super 25 Girls Poll
Wednesday, January 15th, 2014, 10:11am
ATLANTA -- The latest edition of the USA Today Metro Atlanta Super 25 Girls Basketball Poll is here.
There are some new faces this week, and a new number one team.
The McEachern girls take over the top spot, with Southwest Dekalb second. Archer, Parkview and Redan make up the top 5.
Tucker is next, with North Forsyth, Hillgrove, Wesleyan and Norcross completing the top 10.
The Metro Atlanta Super 25 poll is computer generated and produced exclusively for 11Alive News and 11Alive.com by USA Today High School Sports.
ATLANTA -- The latest edition of the USA Today Metro Atlanta Super 25 Boys Basketball Poll is ready for inspection.
A past state champion joins our list this week, but the name at the top stays the same as last week.
The Spartans of Greater Atlanta Christian School stay unbeaten and on top. Allatoona is next, with Columbia and Wheeler in third and fourth. North Gwinnett is fifth headed into a region showdown with Norcross.
Tucker, Miller Grove, Norcross, Milton and Berkmar round out the top 10.
Past state champion South Atlanta joins the poll this week at number 23.
The Metro Atlanta Super 25 poll is computer generated and produced exclusively for 11Alive News and 11Alive.com by USA Today High School Sports
'Rolling quorum' helps local governments skirt open meetings laws
Submitted by Doug Richards, 11Alive Reporter
(WXIA) -- When the powerful meet in secret, it makes it doubly difficult for the public to hold them accountable. One method they're using is what's called a rolling quorum -- where members of a city council or county commission meet -- but one or members listen from another room.
The state open meeting laws explicitly require that meetings of city councils and county commissions give notice of their meetings. But when members don't want the public there -- here's one way around it.
Soldier returns home and makes 2 surprise stops at schools
Monday, January 13th, 2014, 6:32pm
GWINNETT COUNTY, Ga. -- For Army Capt. Robert Abrams, after a year of dedication to country, he can now give way to devotion to family.
Monday he returned from his second tour overseas, and showed up at Creekland Middle School to surprise his 6th grade daughter, Lexi.
She thought everyone gathered to hear her class read poems about military service. She was up first, when her dad walked in.
Before taking his daughter home to celebrate, they made another surprise stop, this time picking up Abrams' 7-year-old son Preston.
Path2College launches savings sweepstakes
Friday, January 10th, 2014, 3:00pm
(WXIA) -- Path2College, Georgia's 529 savings plan, has kicked off this year's $25,000 Reasons to Save Sweepstakes.
The promotion is designed to get parents thinking about saving for college while their children are still young.
Through the sweepstakes, one lucky child will be awarded $15,000 for his or her college education. The winner's school will receive $10,000.
"All you have to do is go online and enter for a chance to win," Path2College exeuctive director Mitch Seabaugh said. "There's nothing to lose by entering, and both a child and his or her school have a lot to gain."
To enter, visit path2college529.com. The sweepstakes is open to current Path2College 529 Plan account owners or non-account owners who live in Georgia and are at least 21 years old.
The contest closes on Sunday, Aug. 10.
Man killed in Snellville house fire
Friday, January 10th, 2014, 12:22pm
SNELLVILLE, Ga. -- A man was killed in a fire that broke out in the Gwinnett County house where he was staying.
The fire started at around 1:45 a.m. Friday at a house in the 800 block of Lake Drive SW in Snellville.
Gwinnett County Fire Capt. Tommy Rutledge said the single-story wood house was fully involved when crews arrived. The flames were moving very close to the house next door as well as live electrical lines overhead.
Three people were inside the house. The man and woman who owned the home suffered minor injuries but managed to escape; a relative who was sleeping in their basement did not.
The victim's body was found at around 8:30 a.m. Rutledge said firefighters discovered the remains while digging through debris.
No names have been released. The body has been turned over to the Gwinnett County Medical Examiner.
The cause of the fire is under investigation.
Ga.'s largest movie studio could open by summer
Submitted by Atlanta Busines... on Friday, January 10th, 8:34 am
GWINNETT COUNTY, Ga. -- Jacoby Development says it could have Georgia's largest movie studio open by this summer in Gwinnett County.
PHOTOS | Movies filmed in the Atlanta area
Company President Scott Condra told Gwinnett Village Community Improvement District leaders on Thursday that the first phase of the $1 billion investment -- opening up seven sound stages in a building once used to manufacture fiber optic cables -- could be complete three to six months after the April closing with OFS.
The project, at the OFS site along Interstate 85, has the potential to transform the Jimmy Carter Boulevard corridor.

redpajama_set_name: RedPajamaCommonCrawl


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Most intakes are not California CARB legal. Check with your local smog laws.


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Smokeless Cigarettes Healthy Or Health Hazard
www.gather.com/viewArticle.action?articleId=2814... - Details
GrantDunn: Good share here... Smokeless Cigarettes Healthy Or Health Hazard
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Complete assortment of Electronic Cigarette accessories
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Having served for ten years active duty, including combat, in the United States Navy did not make me eligible for a pension or stipend of any sort. It did not allow for tax free shopping privileges at the on-base Commissary for groceries or the Navy Exchange for sundry goods. It did not provide for medical or dental benefits. It did not even permit access to the Naval Air Station to show my son the few buildings still standing where I used to work.
All that remained to show for a decade of service to my country were boxed up medals, plaques, awards, memories, and the only perk that honestly lasts forever-friendships tempered by the steely combat of a long-misplaced youth.
Eight months ago, I had a once in a lifetime opportunity to turn back the clock a quarter century, if only temporarily. And if only in fantasy. I had a chance to prowl through decks and levels, over knee-knockers, through passageways and up and down ladders, to enter ship's spaces not seen in more than twenty years. It was in these rooms that the seeds of intense first thoughts on my own mortality were planted. I would step aboard U.S.S. Ranger a final time before she left Puget Sound Navy Shipyard without flags or fanfare, absent the traditional white uniformed sailors manning her rails.
Ranger was being prepared to be towed away, with boilers silent and cold under the perpetual darkness of a forgotten warrior. Ranger was being prepared to be rendered for scrap. For the morning, though, she was ours to share with past comrades in arms, to temporarily join physical reality with memory.
One day and two dozen years earlier, I had launched from Ranger's deck in the predawn Persian Gulf blackness into the unknowns of first combat. In certain ways, it marked the launch of the rest of my life-I was not the same person after that morning. But that was past, worthy of reflection only when surrounded by old squadron-mates, and not as the topic of a moribund séance in a solitary mind. To have this opportunity presented at such a time in my life bared its teeth at coincidence, challenged life with the stubborn insistence of a young man's-and an old salt of a warship's-denial of fate.
Ross Wilhelm, a VA-145 B/N and friend, had called me on the Monday before the tour with the Ranger invitation. The U.S.S. Ranger was decommissioned in 1993. She had fought off the blow torches and junk yard cranes for 22 years. Why scrap her now?
The more I thought about it, though, the more appropriate it seemed to be; everything goes away, everybody dies-that's just the way things are. Maybe it was better to grapple with this fact in the close combat of reality than to push it off until it could no longer be ignored; before the facts of life intercepted a tired mind's fantasies in an ambush of truth.
Tugg Thompson was one of those old friends traveling to Bremerton to say goodbye, a friend who fortunately had retired from the navy and still had an identification card and access to navy facilities. We had both been pilots when on active duty, but I would succumb today and sit in the right seat of his silver Accord and let him do the driving onto the navy base.
My son, Jared, sat in back, skipping school for a lesson in history from has-been shipmates, given on a ship that hadn't sailed in decades. As we drove onto the Keystone Ferry to travel from Whidbey Island to Port Townsend, Tugg's descriptions of past victories and foibles were unrelenting in their energy; he was a tenacious energizer bunny with a heart as big as his enthusiasm for flight, with every word threatening to spin out of control with a fiery clap of his hands. It would be a lesson for Jared not available in a dozen weeks or years of school.
The ferry rolled in the unseen swell of a wintry, still-dark, pre-dawn, priming our memories and arming our resolve. Once in Port Townsend, it would be a one hour drive to the Bremerton Naval Shipyard, where Ross Wilhelm would meet us for the tour. We would need to be off the Ranger ninety minutes later, as she was slated to leave Bremerton in four weeks for the tow around the Cape of Magellan, and then north to Brownsville, Texas. For a ship to die on dry ground took a great deal of preparation indeed.
A half dozen other squadron mates had opted to regret the invite, stating honestly-if imperfectly-that the event would just be too sad.
This I could not fully understand. I had a vague sense of how a soft melancholy might threaten, but to have tales of the past intrude on the present, to manifest themselves through real, physical emotion?
No. To me, it was a celebration of a long-closed chapter, one so distant through the ravages of time, so alien to today's reality, that it was difficult to quite believe that the memories were real.
Would it have been better to have Ranger slowly rust away pier-side without urgency or reason? Would it have been happier to know that Ranger, bereft of visitors or mission, would slowly flake into obscurity? Wasn't the scrap pile just the sort of tidy closure that so many seemed to be searching for in life? Wasn't this cause to celebrate the past?
My mood was far from sad when we got to Ranger. Walking her passageways was energizing as I eagerly peered into each darkened space for a glimpse of the familiar. The view was not disappointing-it was as if movers had come for her furniture and wall hangings, but left behind everything else unmolested.
Walking the passages, which had always been bare, it looked the same as walking onto the darkened ship after a night of liberty in port: mostly quiet, but with the jarring yells of returned revelers always threatening.
Or, was it more akin to the walk from Mid-rats to the stateroom in the middle of the rolling night, standing the alert 15, heavy flight gear hanging loosely on an uncaring young frame, prepared and eager for the urgency of a surprise launch: something vitally important, a mission. It was both. And it was neither. It was real.
I left Ranger seeing and sharing with Jared far more than expected and feeling pretty damn good, without a hint of sadness. I asked Tugg on the return drive how he felt and he agreed. Memories, as important as they are, must be left in the past. Not only is there no choice in the matter, but if allowed to flow with life's natural energy, it is better.
Beauty effortlessly comes in many forms; the challenge is to accept beauty on her own terms. Old friendships work that way.


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Cedar Phytochemicals
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Antimalarial drug confirmed effective on Covid-19
BEIJING — Chinese experts, based on the result of clinical trials, have confirmed that Chloroquine Phosphate, an antimalarial drug, has a certain curative effect on the coronavirus disease (Covid-19), a Chinese official said here Monday.
The experts have "unanimously" suggested the drug be included in the next version of the treatment guidelines and applied in wider clinical trials as soon as possible, Sun Yanrong, deputy head of the China National Center for Biotechnology Development under the Ministry of Science and Technology (MOST), said at a press conference.
Chloroquine Phosphate, which has been used for more than 70 years, was selected from tens of thousands of existing drugs after multiple rounds of screening, Sun said.
According to her, the drug has been under clinical trials in over 10 hospitals in Beijing, as well as in south China's Guangdong Province and central China's Hunan Province, and has shown fairly good efficacy.
In the trials, the groups of patients who had taken the drug have shown better indicators than their parallel groups, in statement of fever, improvement of CT images of lungs, the percentage of patients who became negative in viral nucleic acid tests and the time they need to do so, she said.
Patients taking the drug also take a shorter time to recover, she added.
Sun gave an example of a 54-year-old patient in Beijing, who was admitted to hospital four days after showing symptoms. After taking the drug for a week, he saw all indicators improve and the nucleic acid turn negative.
So far, no obvious serious adverse reactions related to the drug have been found among the over 100 patients enrolled in the clinical trials, she said.
On Feb. 15, several departments including the MOST, the National Health Commission and the National Medical Products Administration called a video conference to listen to drug research and clinical experts' opinions on the drug's efficacy on Covid-19.
The expert team, led by Zhong Nanshan, a renowned respiratory specialist and an academician of the Chinese Academy of Engineering, agreed that Chloroquine Phosphate can be used to treat more Covid-19 patients, Sun said.
Previous in vitro experiments showed that it can block virus infections by changing the acidity and basicity value inside the cell and interfering receptors of SARS coronavirus.
It also shows immune-modulating activity, which may enhance its antiviral effect in vivo and is widely distributed in the whole body, including the lungs, after oral administration.
WHO, China begin field inspections
A joint expert team consisting of experts from China and the World Health Organization on Monday began field inspections on the prevention and control of the Covid-19.
The expert team will go to Beijing, Guangdong Province and Sichuan Province to conduct inspections, said Mi Feng, spokesperson for the National Health Commission (NHC), at a press conference on Monday.
A symposium was held by the NHC Sunday, which was attended by nearly 80 people, including joint expert team members and representatives from the State Council's joint prevention and control mechanism, Mi said.
At the symposium, NHC deputy director Li Bin briefed attendees on nationwide prevention and control measures, vowed to work with the international community to address the challenge of the epidemic and welcomed suggestions from the expert team, according to Mi.
Representatives from State Council ministries and administrations introduced their prevention and control work. The attendees talked with Hubei, the virus-hit province, during a teleconference, and discussed the epidemic situation, control and prevention measures in communities and rural areas, wildlife management and drug and vaccine development.
The joint expert team acknowledged China's prevention and control measures, as well as the dedication of Chinese medical workers, Mi said. (Xinhua)
The mayor of Bugliano, Italy has responded to the viral outbreak by prohibiting orgies inside the city limits.
help fight the new coronavirus?
Doctor obstetrician-gynecologist Marina Bragina. Dnipro (Dnepropetrovsk), Ukraine
Multi-core servers or why alcoholics get drunk. From the point of view of an IT specialist
Treatment of cellulite and local fat deposits. Dnipro, Ukraine.
Cosmetology Dnipro / Dnepropetrovsk: Artichoke Extract; L-canritin; Melilotus extract plus rutin
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This look is so pretty, whilst still maintaining a fashion week edge with pumps and luxe layering. Pink is a huge upcoming colour trend, which the fashion pack are pairing with red and mulberry tones, as seen here.
I love this Lauren Santo Domingo look because it's practical. I know that's not a fashion answer, but look how cosy; she hasn't shrugged her coat off her shoulders, it's all fastened up and it meets the top of her boots! Girl after my own heart. However, she still looks impossibly chic, and shearling is a current trend box she's ticked.
I love this laid-back look. Layered sweaters and ankle boots are winter essentials to most of us, but the little details make this look; a simple low ponytail, ripped jeans over fishnets and small box bags are all major at the moment.
She had to be in this list somewhere; Olivia Palermo, street style royalty. True to her style, she is clashing textures and prints, and it totally works. Her delicate silk neck scarf is making me want to run into the Northern Quarter and buy a vintage scarf ASAP.
I'm not sure how suited this is to the -6 degree streets of New York, but she looks fabulous never the less – and her outfit features the accessory of AW16, the now iconic Gucci belt. This ensemble (minus the fur) is giving me major Billie Holiday vibes and is a look I think I'll bank, to recreate in the warmer months.


Can a third party affect workers' compensation?
Workers' compensation is a useful form of monetary compensation to aid employees that are harmed in the workplace after suffering from an accident. If they are injured in a workplace accident, they have the ability to collect workers' compensation to aid with their medical bills. For these cases, neither the employee or the employer are found responsible for the accident. There is not fault involved in workplace accidents when workers' compensation is collected. This can help keep the relationship between the employee and the employer civil. The employee will not have to file a lawsuit against their employer to collect damages to aid with their medical bills. Instead, they can simply file for workers' compensation to collect compensation. When employees collect workers' compensation, they cannot file a lawsuit against their employer.
Third parties can be involved in workplace accidents. They may have provided some sort of equipment that the employees were using at the time when the accident occurred. If their involvement somehow contributed to the accident that the employee suffered from, they can be held accountable. Employees can file a lawsuit against the third party. Employees are able to collect workers' compensation in the meantime. However, if they win damages from their lawsuit against the third party, they may have to pay back the workers' compensation that they previously collected.
What are the benefits that are included?
In order to help those that are injured in the workplace, workers' compensation is a viable option. This was created to ensure that employees could not get fired right after going through an accident in the workplace that made them no longer able to continue their duties. Through workers' compensation, you may be entitled to monetary compensation. This compensation may include economic damages can include compensation for medical bills, lost wages and lost future wages. Individuals who have been injured due to a workplace accident should seek medical attention to make sure they are in good health and to document their injuries. This can prove the extent of the injuries to collect the proper amount of workers' compensation. Through workers' compensation, victims of an accident may also be entitled to benefits including medical care, supplemental benefits, social security benefits and death benefits, along with the above cash benefits. If an employee is not able to work anymore, they may be entitled to long-term benefits that can help them sustain their quality of life.
How do mass transit accidents occur?
When can I collect workers' compensation?


it. So I went with Elder Beal and Young the next morning (2 AM) to try and send Elder Beal off, and we waited for another 5 hours, and no bus. So I got next to no sleep that night, and ended up sleeping until 2 PM that day. We didn't get a lot done that day, but Elder Young and I went to go visit a few of the people out in our area. The next day, Elder Beal came with us and we went to the area called 'Controle' which is way the heck out in the middle of nowhere. So We did a lot of walking and not a terrible lot of teaching.
But I started feeling sick that day, and my stomach was acting up a lot. We got home, and I basically spent the next two days in the bathroom. I'm feeling slightly better today, but not a ridiculously amount more. So hopefully it will pass.
Not a lot to talk about this week, but I hope to have more this next week!


The Smart Home Co. Design Page — The Smart Home Co.
Home What is a Smart Home?
Following a thorough consultation we will design a bespoke smart home system that is tailored exactly to your needs.
Good design is the foundation of a successful project.
It's about understanding your needs and lifestyle habits and creating solutions to enhance and enrich them. It's about knowing how different technologies work together and about achieving the right aesthetic. Nothing should be an afterthought.
Following an in-depth consultation we'll create a bespoke and detailed system proposal for your project. We'll then review everything with you before you commit, so you'll know exactly what you're getting and what the system can do for you, both now and in the future.
Let us take care of the complexities whilst you enjoy a beautiful, simple and immersive smart home.


Draped in black patent leather with giant wings extending from behind the ankle and heel region is the "Blackout" Jeremy Scott X adidas Originals Wings 2.0. The followup to Scotts marquee model is now available on Adidas website for purchase. Will you sneakerheads be looking to add these murdered out JS Wings to your collection? hit us up and let us know.


Whether you are striving to create the instance for a content strategy to your own manager or wish to start guest blogging in your favourite websites, locating and working with a fantastic publisher is among the most reliable issues it's possible to do in order to enhance your writing abilities. Inside this blog post, we'll appear at techniques to enrich your written communication. In this particular post, I'm going to permit you to improve your writing a lot. Meanwhile, I've got a few quick methods by which it is possible to enhance your writing immediately. When learning an extra language you got to DETECT and REPETITION. The SAT graders need a remarkably special fashion of essay, with a particular sort of format. Many students should have searched the information jointly with the crucial phrases techniques to increase your writing skills for their documents. Training SEATED essays is, in addition, beneficial.
You'll be able to use shapes or other placeholders to signify the pictures you'll eventually employ.
Dont be confounded since the moment you obtain to understand the objective of this essay each of your confusions may shortly be eliminated and you'll be in your way to write a great article along the way to Enhance My Skills. It is also an extraordinary theory to enter the practice of summarizing the structures of your own documents prior to starting writing. I wish to boost my composition writing. Put simply, outstanding composing routines are vital. It is time to quit procrastinating and begin to perform difficult to boost your academic writing. Below you will find a menu of essential composing habits that may benefit your composing abilities. Fortunately you're not alone within the pursuit to enhance your educational writing.
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The SAT Writing section is overflowing with examples similar to this. It does not have to be this agonizing. I'll accentuate my self awareness by writing. That is the greatest way for you to genuinely enrich your Writing rating. Those will enrich my British by creating that will assist me get self-awareness. I am going to enrich my Language so as to adopt lifelong learning. It is vital that you simply master all these as a way to be a fantastic Language learner. Studying how to be an incredible author in English takes plenty of time plus training.
Two easy creases to suit into an envelope are enough.
Many people might not find it vital to write down what their teacher says, but too little attention often leads to scarcity of development. There are a lot of people approaches to foster your degree of English, but only you will discover the best method for you. It is an excellent method to begin discussions as well as to socialize online with whom you may exercise your English more. Allow it to be simple that folks read work. If you really enjoy to write, the function is going to be entertaining sometimes. Most prolific writers appear to experience an all- organic talent. Pupils will have the ability to study from such types of writers by monitoring their function. Not just are you currently heading to capture far better audience curiosity by doing this, but you will accentuate your viewers' ability to remember what you've written. Everybody has an alternative motive why they should boost their authorship.
When each section is completed by you, proofread it.
There's just one strategy to turn into a better writer, and that's through plenty of practice. Take relaxation within the knowledge this is not how creating works.


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The deadliest type of skin cancer is melanoma, which is known to resist chemotherapy. Medical Express states that almost 25% of melanoma is "is driven by oncogenic mutations in the NRAS gene," which means NRAS can be targeted by therapeutic procedures. Unfortunately, for decades, no therapies used to target NRAS have been successful.
In a first, an international team of researchers led by a team of scientists from Boston University found a new activator of NRAS. Moreover, they also developed an inhibitor that will prevent the growth of NRAS mutant melanoma. The findings recently published in the journal Cell provide a new option to treat aggressive skin cancer.
Given the pivotal role of NRAS signaling … and the prominent role of STK19 in NRAS activation, targeting STK19 would represent a potential new therapeutic strategy in melanoma, especially in those with NRAS mutations.
After selecting and testing STK19, the team has also created a small molecule that targets STK19, which is called ZT-012-037-1 (1a). Results showed that the molecule was successful in inhibiting both tumor development and mutant NRAS-STK19 which leads to melanoma cell formation. The tests were replicated in mouse models, which ended with prolonging their lives.
We hope our findings ultimately will be clinically translated into improved care for cancer patients.