Metric Space Theory

Abstract

A topological space is paracompact if every open cover has a locally finite refinement. We present Mary Ellen Rudin’s proof of Stone’s theorem, which asserts that metric spaces are paracompact. A collection of continuous nonnegative functions is a partition of unity subordinate to a given open cover if the sum of the functions is identically unity and the support of each function is contained in an element of the open cover. Urysohn’s lemma is proved and used to show that any open cover of a normal space has a partition of unity subordinate to it. A topological vector space is a vector space with a topology such that vector addition and scalar multiplication are continuous. A TVS is a Banach space if its topology is derived from a norm and it is complete (every Cauchy sequence has a limit). A Banach space is a Hilbert space if its norm is derived from an inner product. A metric space is homeomorphic to a relatively closed subset of a convex subset of a Banach space. A separable metric space can be embedded in a Hilbert space. A TVS is locally convex if every neighborhood of the origin contains a convex neighborhood. Dugundji’s theorem asserts that a continuous function from a closed subset of a metric space to a locally convex TVS has a continuous extension to the entire space.