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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