Open and other kinds of extensions over local compactifications

TL;DR

This paper generalizes classical theorems on local compactifications, characterizing extensions of Tychonoff spaces and providing conditions for various types of extensions such as open, skeletal, and perfect.

Contribution

It describes the set of all locally compact Hausdorff extensions of a Tychonoff space and establishes conditions for maps to have extensions of specific types over these compactifications.

Findings

Characterization of the partially ordered set of all local compactifications

Necessary and sufficient conditions for extensions of maps

Generalization of results by V. Z. Poljakov

Abstract

Generalizing de Vries Compactification Theorem and strengthening Leader Local Compactification Theorem, we describe the partially ordered set $(\LL(X),\le)$ of all (up to equivalence) locally compact Hausdorff extensions of a Tychonoff space $X$. Using this description, we find the necessary and sufficient conditions which has to satisfy a map between two Tychonoff spaces in order to have some kind of extension over arbitrary given in advance Hausdorff local compactifications of these spaces; we regard the following kinds of extensions: open, quasi-open, skeletal, perfect, injective, surjective. In this way we generalize some results of V. Z. Poljakov.