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


54E15 Uniform structures and generalizations
Full Text: EuDML


[1] Frolík Z.: lntroduction to Seminar Uniform Spaces. 1976-77, Math. Inst. ČSAV, Prague.
[2] Frolík Z.: Four functors into paved spaces. Seminar Uniform Spaces 1973-74 directed by Z. Frolík, MÚ ČSAV, Prague, pp. 27-72. · Zbl 0369.54012
[3] Frolík Z.: Locally e-fine measurable spaces. Trans. A.M.S. 196 (1974), pp. 237-247. · Zbl 0297.54023
[4] Ginshurg S., Isbell J. R.: Some operators on uniform spaces. Trans. A.M.S. 93 (1959), pp. 145-168. · Zbl 0087.37601
[5] Isbell J. R.: Uniform spaces. Math. Surveys (12), 1964, A.M.S. · Zbl 0124.15601
[6] Pelant J.: Remark on locally fine spaces. Comment. Math. Univ. Carolinae 16 (1975), pp. 501-504. · Zbl 0316.54028
[7] Pelant J., Preiss D., Vilimovsky J.: On local uniformities. Gen. Top. and its Appl. 8 (1978), pp. 67-71. · Zbl 0397.54022
[8] Pelant J., Rice M. D.: Remarks on e-locally fine spaces. Seminar Uniform Spaces 1976-77 directed by Z. Frolík, Math. Inst. ČSAV, Prague, pp. 51 - 62.
[9] Rice M. D.: Complete uniform spaces. Springcr-Verlag Lecture Notes 378, 400-418. · Zbl 0327.54025
[10] Shchepin E. V.: Topologies of limit spaces. (Russian), Uspechi mat. nauk XXXI, 5 (191), pp. 191-226. · Zbl 0356.54026
[11] Shchepin E. V.: On topological products, groups, and a new class of spaces more general than metric spaces. Dokl. Akad. Nauk 226 (1976), pp. 527-529. · Zbl 0338.54022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.