Topological extension properties

Abstract:

It is known that if a topological property $\mathcal {P}$ of Tychonoff spaces is closed-hereditary, productive, and possessed by all compact $\mathcal {P}$-regular spaces, then each $\mathcal {P}$-regular space $X$ is a dense subspace of a space ${\gamma _\mathcal {P}}X$ with $\mathcal {P}$ such that if $Y$ has $\mathcal {P}$ and $f:X \to Y$ is continuous, then $f$ extends continuously to ${f^\gamma }:{\gamma _\mathcal {P}}X \to Y$. Such topological properties are called extension properties; ${\gamma _\mathcal {P}}X$ is called the maximal $\mathcal {P}$-extension of $X$. In this paper we study the relationships between pairs of extension properties and their maximal extensions. A basic tool is the concept of $\mathcal {P}$-pseudocompactness, which is studied in detail (a $\mathcal {P}$-regular space $X$ is $\mathcal {P}$-pseudocompact if ${\gamma _\mathcal {P}}X$ is compact). A classification of extension properties is attempted, and several means of constructing extension properties are studied. A number of examples are considered in detail.