Journal of the
Korean Mathematical Society JKMS

In this paper, we introduce some new properties of a topological space which are respectively generalizations of Fr\'echet-Urysohn property. We show that countably AP property is a sufficient condition for a space being countable tightness, sequential, weakly first countable and symmetrizable to be ACP, Fr\'echet-Urysohn, first countable and semi-metrizable, respectively. We also prove that countable compactness is a sufficient condition for a countably AP space to be countably Fr\'echet-Urysohn. We then show that a countably compact space satisfying one of the properties mentioned here is sequentially compact. And we show that a countably compact and countably AP space is maximal countably compact if and only if it is Fr\'echet-Urysohn. We finally obtain a sufficient condition for the ACP closure operator $[\cdot]_{ACP}$ to be a Kuratowski topological closure operator and related results.

Keywords: Frechet-Urysohn, sequential, countably Frechet-Urysohn, countable tightness, AP, countably AP, WAP, ACP, WACP, countably compact, and sequentially compact

MSC numbers: 54A20, 54D20, 54D55, 54E25