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Keywords:
AP; WAP; door; submaximal; nodec; unique sequential limit
AP; WAP; door; submaximal; nodec; unique sequential limit
Summary:
The definitions of AP and WAP were originated in categorical topology by A. Pultr and A. Tozzi, Equationally closed subframes and representation of quotient spaces, Cahiers Topologie Géom. Différentielle Catég. 34 (1993), no. 3, 167-183. In general, we have the implications: $T_2\Rightarrow KC \Rightarrow US \Rightarrow T_1$, where $KC$ is defined as the property that every compact subset is closed and $US$ is defined as the property that every convergent sequence has at most one limit. And a space is called 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 Fréchet-Urysohn and $US$, which is a generalization of Hong's result in On spaces in which compact-like sets are closed, and related spaces, Commun. Korean Math. Soc. 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.
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