Mi Ae Moon, Myung Hyun Cho, Junhui Kim
On AP spaces in concern with compact-like sets and submaximality

Comment.Math.Univ.Carolin. 52,2 (2011) 293-302.

Abstract:The definitions of AP and WAP were originated in categorical topology by A.~Pultr and A.~Tozzi, {\it Equationally closed subframes and representation of quotient spaces\/}, Cahiers Topologie G\'eom. Diff\'erentielle Cat\'eg. {\bf 34} (1993), no.~3, 167--183. In general, we have the implications: $T_2\Rightarrow KC \Rightarrow US \Rightarrow T_1$, where {\it KC\/} is defined as the property that every compact subset is closed and {\it US\/} is defined as the property that every convergent sequence has at most one limit. And a space is called {\it submaximal\/} if every dense subset is open. In this paper, we prove that: (1) every AP $T_1$-space is US, (2) every nodec WAP $T_1$-space is submaximal, (3) every submaximal and collectionwise Hausdorff space is AP. We obtain that, as corollaries, (1) every countably compact (or compact or sequentially compact) AP $T_1$-space is Fr\'echet-Urysohn and US, which is a generalization of Hong's result in {\it On spaces in which compact-like sets are closed, and related spaces\/}, Commun. Korean Math. Soc. {\bf 22} (2007), no.~2, 297--303, (2) if a space is nodec and $T_3$, then submaximality, AP and WAP are equivalent. Finally, we prove, by giving several counterexamples, that (1) in the statement that every submaximal $T_3$-space is AP, the condition $T_3$ is necessary and (2) there is no implication between nodec and WAP.

Keywords: AP, WAP, door, submaximal, nodec, unique sequential limit
AMS Subject Classification: 54D10 54D55

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