Irreducible Quotient Maps From Locally Compact Separable Metric Spaces

TL;DR

This paper characterizes when a Hausdorff quotient of a locally compact separable metric space can be obtained via an irreducible quotient map, extending classical theorems to a broader class of spaces.

Contribution

It provides new equivalences and conditions for irreducible quotient maps from standard spaces, extending classical results to locally compact spaces.

Findings

Equivalence of three conditions for irreducible quotient maps

Extension of Whyburn and Zarikian theorems to locally compact spaces

New results for quotients of locally compact subsets of the real line

Abstract

Let X be a Hausdorff quotient of a standard space (that is of a locally compact separable metric space). It is shown that the following are equivalent: (i) X is the image of an irreducible quotient map from a standard space; (ii) X has a sequentially dense subset satisfying two technical conditions involving double sequences; (iii) whenever q : Y\to X is a quotient map from a standard space Y , the restriction q_{\st}|V is an irreducible quotient map from V onto X (where q_{\st} : Y_{\st}\to X is the pure quotient derived from q, and V is the closure of the set of singleton fibres of Y_{\st}). The proof uses extensions of the theorems of Whyburn and Zarikian from compact to locally compact standard spaces. The results are new even for quotients of locally compact subsets of the real line.