Koushesh, M. (2011). One-point extensions of locally compact paracompact spaces. Bulletin of the Iranian Mathematical Society, 37(No. 4), 199-228.

M. R. Koushesh. "One-point extensions of locally compact paracompact spaces". Bulletin of the Iranian Mathematical Society, 37, No. 4, 2011, 199-228.

Koushesh, M. (2011). 'One-point extensions of locally compact paracompact spaces', Bulletin of the Iranian Mathematical Society, 37(No. 4), pp. 199-228.

Koushesh, M. One-point extensions of locally compact paracompact spaces. Bulletin of the Iranian Mathematical Society, 2011; 37(No. 4): 199-228.

One-point extensions of locally compact paracompact spaces

Receive Date: 16 March 2010,
Revise Date: 21 July 2010,
Accept Date: 21 July 2010

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 $Y'$ of $X$ let $Yleq Y'$, 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 $Yackslash X$ is a singleton. An extension $Y$ of $X$ is called {em first-countable}, if $Y$ is first-countable at points of $Yackslash X$. 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 $eta Xackslash X$. Here ${mathcal P}$ is subject to some requirements and include $sigma$-compactness and the Lindel"{o}f property as special cases.