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Functions having closed graphs. (English) Zbl 0806.54011
A function \(f:X \to Y\) is termed \(c\)-continuous (resp. \(c^*\)- continuous) if for each \(x \in X\) and each open neighborhood \(V\) of \(f(x)\) having compact (resp. countably compact) complement, there exists an open neighborhood \(U\) of \(x\), such that \(f(U) \subset V\). The authors provide characterizations of the local compactness and local countable compactness in terms of \(c\)-continuous or \(c^*\)-continuous functions having closed graphs. Also conditions are given for the set of functions with closed graphs to be a closed subset of the space of all functions with the topology of uniform convergence or the topology of uniform convergence on compact sets. An answer to a question of P. E. Long and M. D. Hendrix on \(c\)-continuous functions is given.

MSC:
54C08 Weak and generalized continuity
54C10 Special maps on topological spaces (open, closed, perfect, etc.)
54C35 Function spaces in general topology
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