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On minimal Hausdorff and minimal Urysohn functions. (English) Zbl 1252.54013

The authors develop a new characterization of minimal Hausdorff functions and introduce the concept of minimal Urysohn functions. They also investigate hereditary and productive properties for both minimal Hausdorff and Urysohn functions.
A Hausdorff space \(X\) is minimal Hausdorff if it has no strictly coarser Hausdorff topology. A continuous function \(f:X\to Y\) is said to be Hausdorff (respectively, Urysohn) if for every \(x,y\in X\) such that \(x\not=y\) and \(f(x)=f(y)\) there are two neighbourhoods \(U_x\) and \(V_x\) of \(x\) and \(y\) such that \(U_x \bigcap V_y = \emptyset\) (respectively, \(cl_X(U_x) \bigcap cl_X(V_y)=\emptyset\)). A Hausdorff function \(f:(X, \tau(X))\to Y\) is minimal Hausdorff (respectively, minimal Urysohn) if for any topology \(\sigma\) on \(X\), \(\sigma \not\subset \tau(X)\), \(f^{\prime}:(X,\sigma)\to Y\), \(x\to f(x)\), is not Hausdorff (respectively, Urysohn).

MSC:

54C10 Special maps on topological spaces (open, closed, perfect, etc.)
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
54C20 Extension of maps
54D10 Lower separation axioms (\(T_0\)–\(T_3\), etc.)
54D25 “\(P\)-minimal” and “\(P\)-closed” spaces
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References:

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