Compactification-like extensions

Abstract

Let $X$ be a space. A space $Y$ is called an extension of $X$ if $Y$ contains $X$ as a dense subspace. For an extension $Y$ of $X$ the subspace $Y\backslash X$ of $Y$ is called the remainder of $Y$. Two extensions of $X$ are said to be equivalent if there is a homeomorphism between them which fixes $X$ pointwise. For two (equivalence classes of) extensions $Y$ and $Y'$ of $X$ let $Y\leq Y'$ if there is a continuous mapping of $Y'$ into $Y$ which fixes $X$ pointwise. Let $P$ be a topological property. An extension $Y$ of $X$ is called a $P$-extension of $X$ if it has $P$. If $P$ is compactness then $P$-extensions are called ompactifications. The aim of this article is to introduce and study classes of extensions (which we call compactification-like $P$-extensions, where $P$ is a topological property subject to some mild requirements) which resemble the classes of compactifications of locally compact spaces. We formally define compactification-like $P$-extensions and derive some of their basic properties, and in the case when the remainders are countable, we characterize spaces having such extensions. We will then consider the classes of compactification-like $P$-extensions as partially ordered sets. This consideration leads to some interesting results which characterize compactification-like $P$-extensions of a space among all its Tychonoff $P$-extensions with compact remainder. Furthermore, we study the relations between the order-structure of classes of compactification-like $P$-extensions of a Tychonoff space $X$ and the topology of a certain subspace of its outgrowth $\beta X\backslash X$. We conclude with some applications, including an answer to an old question of S. Mr\'{o}wka and J.H. Tsai: For what pairs of topological properties $P$ and $Q$ is it true that every locally-$P$ space with $Q$ has a one-point extension with both $P$ and $Q$?