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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.

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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