F-supercontinuous functions. (English) Zbl 1189.54013

The authors introduce \(F\)-supercontinuity, a strong variant of continuity, relate it to other notions of continuity and discuss its topological relevance.
Let \(X\) and \(Y\) be topological spaces. The authors say that a function \(f : X \rightarrow Y\) is \(F\)-supercontinuous if for every \(x \in X\) and open set \(U \subseteq Y\) containing \(f (x)\) there exists an open set \(V \subseteq X\) containing \(x\) such that \(V\) can be written as a union of zero sets and \(f (V) \subseteq U\).
In the paper, this notion of continuity is related to several other variants of continuity. Characterizations of \(F\)-supercontinuity are given in terms of suitable mapping and convergence properties. The authors introduce \(F\)-open and \(F\)-closed sets and functions, \(F\)-compactness, \(F\)-quotient topologies and maps, and discuss the interplay between these notions. The graphs, the compositions and the topological products of \(F\)-continuous functions are also investigated.


54C08 Weak and generalized continuity
54C10 Special maps on topological spaces (open, closed, perfect, etc.)
54D10 Lower separation axioms (\(T_0\)–\(T_3\), etc.)
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54D30 Compactness
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