On sequentially closed subsets of the real line in

Abstract

We show: iff every countable product of sequential metric spaces (sequentially closed subsets are closed) is a sequential metric space iff every complete metric space is Cantor complete. Every infinite subset X of has a countably infinite subset iff every infinite sequentially closed subset of includes an infinite closed subset. The statement “ is sequential” is equivalent to each one of the following propositions: Every sequentially closed subset A of includes a countable cofinal subset C, for every sequentially closed subset A of , is a meager subset of , for every sequentially closed subset A of , , every sequentially closed subset of is separable, every sequentially closed subset of is Cantor complete, every complete subspace of is Cantor complete.