  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.
