Compactification-like extensions

TL;DR

This paper introduces and analyzes classes of extensions of topological spaces that generalize compactifications, focusing on properties like countability and their order-structure, with applications including answering longstanding questions in topology.

Contribution

It defines compactification-like $P$-extensions, characterizes spaces with such extensions, and explores their order-structure and relation to the topology of the space's outgrowth.

Findings

Characterization of spaces with countable compactification-like $P$-extensions

Order-structure of classes of compactification-like $P$-extensions

Answer to the question of when spaces with properties $P$ and $Q$ admit one-point extensions with both properties.

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 $Q$?