Compact and extremally disconnected spaces (Q1774723)

Summary: Viglino defined a Hausdorff topological space to be \(C\)-compact if each closed subset of the space is an \(H\)-set in the sense of Veličko. We study the class of Hausdorff spaces characterized by the property that each closed subset is an \(S\)-set in the sense of Dickman and Krystock. Such spaces are called \(C\)-\(s\)-compact. Recently, the notion of strongly subclosed relation, introduced by Joseph, has been utilized to characterize \(C\)-compact spaces as those with the property that each function from the space to a Hausdorff space with a strongly subclosed inverse isclosed. Here, it is shown that \(C\)-\(s\)-compact spaces are characterized by the property that each function from the space to a Hausdorff space with a strongly sub-semiclosed inverse is a closed function. It is established that this class of spaces is the same as the class of Hausdorff, compact, and extremally disconnected spaces. The class of \(C\)-\(s\)-compact spaces is properly contained in the class of \(C\)-compact spaces as well as in the class of \(S\)-closed spaces of Thompson. In general, a compact space need not be \(C\)-\(s\)-compact. The product of two \(C\)-\(s\)-compact spaces need not be \(C\)-\(s\)-compact.