Uniform Hyperspaces and Hausdorff Completions

We consider the uniform hyperspaces of a uniform space X, its associated Hausdorff space sX and its Hausdorff completion nX, and investigate the relationships between the functors s and n and the various hyperspace functors. Thus Theorem 1 says that there is a natural isomorphism between the functors TTE and TTETT, where ii'A'is the space of non-empty closed subsets of X, endowed with the Hausdorff uniformity, and this result is applied to a consideration of which spaces have hypercompletions, to the question of compactifications of hyperspaces, and to finding when the Hausdorff uniformity is compatible with the finite (Vietoris) topology. The study is extended in §2 to hyperspaces consisting of compact, precompact, relatively compact and finite sets. §3 contains some applications to linear spaces and in particular to completions of quotient spaces. The final section is an investigation of hyperspaces arising from two uniformities on a set, associated in a certain way. In the course of the paper some generalizations are given of results of the Robertsons in [10] on spaces of compact, " weakly " compact and " weakly " relatively compact subsets. Introduction It is well known that from an arbitrary uniform space X we can construct an associated Hausdorff space sX, and then embed sX in a complete Hausdorff space nX (see e.g. Bourbaki [2]), and that s and n are actually functors on the category UNI, of uniform spaces with uniformly continuous maps, into itself. If rj is the uniformity on X, we shall sometimes denote by srj, nrj the uniformities on sX and nX respectively. If % is any base for rj, the sets of the form U = {(A, B):Ac U(B) and B <= U(A)} where U e °U, form a base for a uniformity fj on the hyperspace SX of non-empty subsets of X, called the Hausdorff uniformity (see [9]). It induces a uniformity on EX, the set of non-empty closed subsets, and also on the sets CX, PX, RX and FX, consisting respectively of the compact closed, precompact closed, relatively compact and finite subsets. We shall also consider the sets C0X and P0X consisting of the compact and precompact subsets respectively. We shall sometimes denote these uniform hyperspaces by (SX, fj), (EX, fj), (CX, fj) etc., or by E(X, rj), or just EX. When there is need to distinguish uniformities we shall use the notation (E(X, rj), fj), (E(X, £), fj). Throughout, the word isomorphism is used to mean uniform isomorphism. If/: X -> Y is a uniformly continuous map between uniform spaces, then we define EfiEX^EY and Sf: SX -• SY by Ef(A)=f(AJ, Sf(A)=f(A) for each A e EX(resp. SX). These maps are uniformly continuous, and it is easy to see that, with the appropriate image morphism, S, E, C, P, R, F, CQ, and Po are all functors UNI -> UNI. Most of the results of this paper will be framed in terms of category theory, using the equality sign between functors to represent" natural isomorphisms ". Received 14 July, 1974; revised 25 June, 1975. [J. LONDON MATH. SOC. (2), 12 (1976), 505-512]