CHAPTER XVI - COMPACT SPACES

Abstract

This chapter focuses on compact spaces. A topological space which is the union of two compact sets is compact. The Cartesian product of compact spaces is a compact space. Every countable open cover contains a finite subcover. Obviously, a compact space is countably compact, while the converse is not true. If, however, the space is supposed to be metric, then compactness and countable compactness are equivalent. It is stated that every countably compact metric space is separable, and hence contains a countable open base. Every countably compact metric space X is compact. Thus, for metric spaces, compactness and countable compactness are equivalent.