A Theorem on Countably Frechet-Urysohn Spaces

Abstract

In this paper we study some properties of countably Frechet-Urysohn spaces and show a sufficient condition, which is different from the result ([3, Proposition 8.7]) of E. A. Michael, for countably Frechet-Urysohn spaces to be Frechet-Urysohn. And then we obtain a condition which is equivalent to being determined by countable subsets for countably Frechet-Urysohn spaces as a corollary. The following are our main results : Let (X;T) be a countably Frechet-Urysohn space. (1) Suppose (X;T) is sequentially compact. Then (X;T) is maximal sequentially compact if and only if (X;T) is Frechet-Urysohn.(2) Suppose (X;T) is non-sequentially compact. Then the one-point sequential compactification (X;T) of (X;T) is maximal sequentially compact if and only if (X;T) isFrechet-Urysohn.