  A space $Y$ is called an {\em extension} of a space $X$ if $Y$ contains $X$
as a dense subspace. Two extensions of $X$ are said to be {\em equivalent} if
there is a homeomorphism between them which fixes $X$ point-wise. For two
(equivalence classes of) extensions $Y$ and $Y'$ of $X$ let $Y\leq Y'$ if there
is a continuous function of $Y'$ into $Y$ which fixes $X$ point-wise. An
extension $Y$ of $X$ is called a {\em one-point extension} if $Y\backslash X$
is a singleton. An extension $Y$ of $X$ is called {\em first-countable} if $Y$
is first-countable at points of $Y\backslash X$. Let ${\mathcal P}$ be a
topological property. An extension $Y$ of $X$ is called a {\em ${\mathcal
P}$-extension} if it has ${\mathcal P}$.
  In this article, for a given locally compact paracompact space $X$, we
consider the two classes of one-point \v{C}ech-complete ${\mathcal
P}$-extensions of $X$ and one-point first-countable locally-${\mathcal P}$
extensions of $X$, and we study their order-structures, by relating them to the
topology of a certain subspace of the outgrowth $\beta X\backslash X$. Here
${\mathcal P}$ is subject to some requirements and include $\sigma$-compactness
and the Lindel\"{o}f property as special cases.

General Topology