The isomorphisms of the category of uniform spaces and related categories

Abstract

In this paper we determine the isomorphisms of the categories of uniform spaces, proximity spaces and all full embeddings of the categories of several classes of topological spaces (corresponding to usual separation axioms) into themselves. In the lemmas we investigate the relation of the forgetful functors to concrete categories and to functors between concrete categories, in a general setting. To finish we mention several unsolved problems. Uniform and proximity spaces are not assumed to be separated. Unif denotes the category of uniform spaces, HUnif denotes that of separated uniform spaces. THEOREM. Every isomorphism i of Unif onto itself is of the following form: for each uniform space X there is an isomorphism ix:X~i(X), and for each map f: X~ Y the following diagram commutes: X f.~Y ix iv i(X) ~(f), i(Y)