Contra-continuous functions and strongly \(S\)-closed spaces.

*(English)*Zbl 0840.54015Summary: 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, LindelĂ¶f, etc.) |

54C08 | Weak and generalized continuity |

54G99 | Peculiar topological spaces |