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Sequentially closed and absolutely closed spaces. (English) Zbl 0654.54016

Let \({\mathcal P}\) be a category of spaces. A \({\mathcal P}\)-space is \({\mathcal P}\)- closed (respectively, sequentially \({\mathcal P}\)-closed, absolutely \({\mathcal P}\)-closed) if X is closed (respectively, sequentially closed, the equalizer of two continuous functions into a \({\mathcal P}\)-space) in every \({\mathcal P}\)-space in which X can be embedded. A space is US (respectively, SUS) if every convergent sequence in X has a unique limit point (respectively, cluster point). A US (respectively, SUS) space X is sequential US-closed (respectively, SUS-closed) iff X is sequentially (respectively, countably) compact. An absolutely US-closed space is sequentially US-closed, but the converse is false; a SUS-space is absolutely SUS-closed iff it is sequentially SUS-closed. Also, an US- closed (respectively, SUS-closed) space is finite (respectively, compact).
Reviewer: J.R.Porter

MSC:

54D25 “\(P\)-minimal” and “\(P\)-closed” spaces
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