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Contra-continuous functions and strongly $$S$$-closed spaces. (English) Zbl 0840.54015
Summary: M. Ganster and I. L. Reilly [ibid. 12, No. 3, 417-424 (1989; Zbl 0676.54014)] introduced and studied the notion of $$LC$$-continuous functions via the concept of locally closed sets. In this paper we consider a stronger form of $$LC$$-continuity called contra-continuity. We call a function $$f: (X,\tau)\to (Y,\sigma)$$ contra-continuous if the preimage of every open set is closed. A space $$(X, \tau)$$ is called strongly $$S$$-closed if it has a finite dense subset or equivalently if every cover of $$(X, \tau)$$ by closed sets has a finite subcover. We prove that contra-continuous images of strongly $$S$$-closed spaces are compact as well as that contra-continuous, $$\beta$$-continuous images of $$S$$-closed spaces are also compact. We show that every strongly $$S$$-closed space satisfies FCC and hence is nearly compact.

##### MSC:
 54C10 Special maps on topological spaces (open, closed, perfect, etc.) 54D20 Noncompact covering properties (paracompact, Lindelöf, etc.) 54C08 Weak and generalized continuity 54G99 Peculiar topological spaces
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