CHAPTER IV - COMPACT SPACES AND RELATED TOPICS

Abstract

This chapter presents an account of advanced theories on compact spaces. It discusses Tychonoff's product theorem. The Hilbert cube is a compact metric space because it is the product of countably many closed segments that are compact. For every completely regular space R, there exists a compact T2-space β(R) such that R is homeomorphic with a subspace R' of β(R), R' is dense in β(R), and every bounded real-valued continuous function over R' can be extended to a continuous function over β(R). It is an interesting problem to find a compact space R* for a given topological space R such that R is homeomorphic with a dense subset R' of R*. Generally, a space such as R* is called a compactification of R. The concept of compactification appears at present in fields other than general topology. There is a tremendous number of extensions and modifications of the concept of compact space. Paracompact and locally compact spaces are significant examples of these extensions and modifications.