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H-closed functions. (English) Zbl 0962.54011

A continuous function \(f:X\to Y\) is Hausdorff if each pair \(a,b\in X\) such that \(a\neq b\), \(a,b\in f^{-1}(y)\) for some \(y\in Y\) can be separated by open subsets of \(X\). A Hausdorff function \(f:X\to Y\) is H-closed if, for each Hausdorff extension \(g:Z\to Y\), \(X\) is closed in \(Z\).
It is shown that every Hausdorff function has an H-closed extension. Perfect and regular functions are characterized among Hausdorff functions. Here regularity of \(f:X\to Y\) means that \(f\) is continuous and if \(F\) is a closed subset of \(X\) and \(x\in X\smallsetminus F\), there is an open neighbourhood \(V\) of \(f(x)\) such that \(x\) and \(F\cap f^{-1}(V)\) can be separated by open subsets of \(f^{-1}(V)\).

MSC:

54C10 Special maps on topological spaces (open, closed, perfect, etc.)
54C20 Extension of maps
54D25 “\(P\)-minimal” and “\(P\)-closed” spaces
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