Components of first-countability and various kinds of pseudoopen mappings (Q617733)

Three subclasses of the well-known class of pseudoopen mappings are introduced (\(\omega\)-pseudoopen, strictly \(\omega\)-pseudoopen, almost \(\omega\)-pseudo\-open mappings). First countability of images of \(T_1\) spaces under these mappings is investigated. Let \(f:X\to Y\) be a continuous mapping from a \(T_1\) space \(X\) onto a regular pseudocompact space \(Y\). Then \(Y\) is first countable in each of the following cases: (1) \(f\) is \(\omega\)-pseudoopen, (2) \(f\) is almost \(\omega\)-pseudoopen, (3) \(X\) is metric and \(f\) is a pseudopen almost \(S\)-mapping (for any non-isolated point \(y\in Y\), \(f^{-1}(y)\) is separable). If \(f\) is strictly \(\omega\)-pseudopen, then \(f\) is biquotient. A Hausdorff topological group which is the image of a \(T_1\)-space under an \(\omega\)-pseudoopen mapping is metrizable.