SOME OPERATORS ON UNIFORM SPACES

Introduction. The principal results in this paper concern a class of uniform spaces defined by the condition that every uniformly locally uniform covering is uniform. Such spaces, called locally fine spaces, generalize the fine spaces, i.e. spaces bearing the finest uniformity consistent with the topology. The most striking results are (i) that, subject to a mild cardinality restriction, the completion of a locally fine space is determined by its uniformly continuous real-valued functions (generalization of Shirota's theorem [14] for fine spaces), and (ii) that every real-valued uniformly continuous function on a subspace of a locally fine space can be extended (generalization of a known result for fine spaces). However, both of these results are trivial for subspaces of fine spaces (i.e. the step from fine spaces to subspaces is trivial), and we have not been able to determine whether this includes all the locally fine spaces. The principal value of the concept 'locally fine" is that it is a simply defined uniform property which distinguishes a well-behaved class resembling the fine spaces (for which no uniform characterization is known). The class is closed under forming subspaces. For each uniform space ,uX there is a coarsest locally fine uniformity X),j finer than A. The operator X commutes with completion and preserves both subspaces and uniformly continuous functions. (That is, if f: ,uX-r* Y is uniformly continuous then so is f: XyXX-*>u Y; ,XuX does not have the same uniformly continuous real-valued functions as pX.) These properties of X are easy to prove once the existence of Xjt is established. The existence proof rests heavily on the lemma, any locally fine uniformity finer than a complete metric uniformity is fine, and on the known results that every uniformity is a union of metric nonseparated uniformities and that every metric space is paracompact. These results are employed also in the generalization of Shirota's theorem. The last section of the paper examines some of the constructions more closely and indicates some more direct proofs. The first half of the paper is a review of the elementary theory of uniform spaces, supplemented with some results not all of which are related. The best of these results is that every countable uniform covering has a countable uniform star-refinement. As a corollary it follows that a uniformity having a