Abstract
In this paper, we introduce some new properties of a topological space which are respectively generalizations of -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, , first countable and semimetrizable, respectively. We also prove that countable compactness is a sufficient condition for a countably AP space to be countably . 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 . We finally obtain a sufficient condition for the ACP closure operator to be a Kuratowski topological closure operator and related results.
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