Weakly closed functions and Hausdorff spaces. (English) Zbl 0622.54008

For \(A\subseteq X\), let \(Cl_{\theta}A=\{x\in X:(Cl U)\cap A\neq \emptyset\) for any open \(U\supseteq \{x\}\}\); A is \(\theta\)-closed if \(A=Cl_{\theta}A\). Two subsets A, B of X are strongly separated if there are open \(U\supseteq A\), \(V\supseteq B\) with (Cl U)\(\cap (Cl V)=\emptyset\). A function f is weakly closed if Cl f(Int C)\(\subseteq f(C)\) for any closed \(C\subseteq X\); *-open if Cl f(U)\(=Cl_{\theta}f(U)\) for any open \(U\subseteq X\); nearly almost open if there is an open basis \({\mathcal B}\) for the topology of its codomain such that \(f^{-1}(Cl V)\subseteq Cl f^{-1}(V)\) for each \(V\in B\); regular closed if f(C) is closed for any regularly closed C. A function f has strongly closed graph if \(y\neq f(x)\) implies there is open \(U\times V\supseteq \{(x,y)\}\) such that (U\(\times Cl V)\cap G(f)=\emptyset.\)
P. E. Long and L. L. Herrington [Rend. Circ. Mat. Palermo, II. Ser. 27, 20-28 (1978; Zbl 0416.54005)] showed that every surjection with strongly closed graph has Hausdorff range. While asserting a more general result, T. Noiri [Math. Nachr. 99, 217-219 (1980; Zbl 0476.54009)] proved that every open surjection with closed graph has Hausdorff range. Both results are generalized by establishing that nearly almost open maps are *-open (but not conversely), and that *-open maps with closed graph have strongly closed graph. An unpublished result by the same authors gives that *-open, weakly closed surjections with \(\theta\)-closed fibres have Hausdorff range. Further, it is shown that every weakly closed (resp., regular closed) surjection for which all pairs of fibres are strongly separated (resp., separated) has Hausdorff range.
Reviewer: D.L.Grant


54C10 Special maps on topological spaces (open, closed, perfect, etc.)
54D10 Lower separation axioms (\(T_0\)–\(T_3\), etc.)
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