Metric Spaces

Abstract

The concept of a metric is defined, some examples are studied, and various techniques for constructing metrics are developed. The neighborhood structure of a metric space is defined, and the notions of weakly continuous and uniformly continuous functions are introduced. Completeness is defined in Section 3, and the construction of the completion is carried through. Following Brouwer, we define a compact space to be a metric space that is complete and totally bounded. Compact and locally compact spaces are studied in Sections 4–6. Constructivizations of various classical results, such as Ascoli’s theorem, the Stone-Weierstrass theorem, and the Tietze extension theorem, are given. The concept of a located set, due to Brouwer, plays an important role. Crucial for later developments is Theorem (4.9), a partial substitute for the classical result that a closed subset of a compact space is compact.