Compactification-like extensions
Tom 476 / 2011
Streszczenie
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 $Y\setminus X$ 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 ${\mathcal P}$ be a topological property. An extension $Y$ of $X$ is called a ${\mathcal P}$-extension of $X$ if it has ${\mathcal P}$. If ${\mathcal P}$ is compactness then ${\mathcal P}$-extensions are called compactifications. The aim of this article is to introduce and study classes of extensions (which we call compactification-like ${\mathcal P}$-extensions, where ${\mathcal P}$ is a topological property subject to some mild requirements) which resemble the classes of compactifications of locally compact spaces. We formally define compactification-like ${\mathcal P}$-extensions and derive some of their basic properties, and in the case when the remainders are countable, we characterize spaces having such extensions. We then consider the classes of compactification-like ${\mathcal P}$-extensions as partially ordered sets. This consideration leads to some interesting results which characterize compactification-like ${\mathcal P}$-extensions of a space among all its Tychonoff ${\mathcal P}$-extensions with compact remainder. Furthermore, we study the relations between the order-structure of classes of compactification-like ${\mathcal P}$-extensions of a Tychonoff space $X$ and the topology of a certain subspace of its outgrowth $\beta X\setminus X$. We conclude with some applications, including an answer to an old question of S. Mrówka and J. H. Tsai: For what pairs of topological properties ${\mathcal P}$ and ${\mathcal Q}$ is it true that every locally-$\mathcal{P}$ space with $\mathcal{Q}$ has a one-point extension with both $\mathcal{P}$ and $\mathcal{Q}$? An open question is raised.
