On Spaces in which compact-like sets are closed, and related spaces

Abstract

In this paper, we study on C-closed spaces, SC-closed spaces andrelated spaces. We show that a sequentially compact SC-closedspace is sequential and as corollaries obtain that a sequentiallycompact space with unique sequential limits is sequential if andonly if it is C-closed [7, 1.19 Proposition] and everysequentially compact SC-closed space is C-closed.We also show that a countably compact WAP and C-closed space issequential and obtain that a countably compact (or compact orsequentially compact) WAP-space with unique sequential limits issequential if and only if it is C-closed as a corollary.Finally we prove that a weakly discretely generated AP-space isC-closed. We then obtain that every countably compact (or compactor sequentially compact) weakly discretely generated AP-space isFr'echet-Urysohn with unique sequential limits, for weaklydiscretely generated AP-spaces, unique sequential limits equivKC equiv C-closed equiv SC-closed, and every continuoussurjective function from a countably compact (or compact orsequentially compact) space onto a weakly discretely generatedAP-space is closed as corollaries.