Irreducible quotient maps from locally compact separable metric spaces

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→X is a quotient map from a standard space Y, the restriction q⁎|V is an irreducible quotient map from V onto X (where q⁎:Y⁎→X is the pure quotient derived from q, and V is the closure of the set of singleton fibres of Y⁎). 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.