Document Type : Research Paper

**Author**

**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.

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.

**Keywords**

December 2011

Pages 199-228

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