Compact and extremally disconnected spaces

Viglino defined a Hausdorff topological space to be C-compact if each closed subset of the space is anH-set in the sense of Velǐcko. In this paper, 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 calledC-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 is closed. 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.