Almost closed sets and topologies they determine

Abstract

summary:We prove that every countably compact AP-space is Fréchet-Urysohn. It is also established that if $X$ is a paracompact space and $C_p(X)$ is AP, then $X$ is a Hurewicz space. We show that every scattered space is WAP and give an example of a hereditarily WAP-space which is not an AP-space.