Irreducible quotient maps from locally compact separable metric spaces

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