On Spaces in which compact-like sets are closed, and related spaces
Commun. Korean Math. Soc. 2007 Vol. 22, No. 2, 297-303
Printed June 1, 2007
Woo Chorl Hong
Pusan National University
Abstract : In this paper, we study on C-closed spaces, SC-closed spaces and related spaces. We show that a sequentially compact SC-closed space is sequential and as corollaries obtain that a sequentially compact space with unique sequential limits is sequential if and only if it is C-closed [7, 1.19 Proposition] and every sequentially compact SC-closed space is C-closed. We also show that a countably compact WAP and C-closed space is sequential and obtain that a countably compact (or compact or sequentially compact) WAP-space with unique sequential limits is sequential if and only if it is C-closed as a corollary. Finally we prove that a weakly discretely generated AP-space is C-closed. We then obtain that every countably compact (or compact or sequentially compact) weakly discretely generated AP-space is Fr\'echet-Urysohn with unique sequential limits, for weakly discretely generated AP-spaces, unique sequential limits $\equiv$ KC $\equiv$ C-closed $\equiv$ SC-closed, and every continuous surjective function from a countably compact (or compact or sequentially compact) space onto a weakly discretely generated AP-space is closed as corollaries.
Keywords : KC, C-closed, SC-closed, Frechet-Urysohn, sequential, AP, WAP, weakly discretely generated
MSC numbers : 54A20, 54A25, 54D20, 54D55, 54D99
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