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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.

##### MSC:
 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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