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Locally fine uniformities and normal covers. (English) Zbl 0656.54020
A fine uniform space is one whose uniformity is the finest uniformity inducing the uniform topology. A (separated) uniform space \((X,U)\) is subfine if \((X,U)\) is a uniform subspace of a fine uniform space and is locally fine if \(U\) equals its own Ginsburg-Isbell derivative. Subfine spaces and locally fine spaces form two coreflective classes of the category of uniform spaces and uniformly continuous mappings and have been studied extensively. In particular, it is known that each subfine uniform space is locally fine. The author proves that each locally fine uniform space is subfine, thus answering affirmatively the previously open question of the coincidence of these two classes of uniform spaces. The proof relies on the notion of special trees (trees in which each chain is finite) and their use in constructing certain refinements of locally finite covers consisting of regular open sets in products of complete metric spaces.
Reviewer: S.C.Carlson

MSC:
54E15 Uniform structures and generalizations
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