One-point extensions of locally compact paracompact spaces

Abstract

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 $Y^{\prime }$ of $X$ let $Y\le Y^{\prime }$ if there is a continuous function of $Y^{\prime }$ 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 Čech-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 Lindelöf property as special cases.

22 pages

References in corpus (1)

• Compactification-like extensions

Cited by in corpus (2)

• Compactification-like extensions
• Topological extensions with compact remainder