Subcategories of uniform spaces

Author:
Michael D. Rice

Journal:
Trans. Amer. Math. Soc. **201** (1975), 305-314

MSC:
Primary 54E15

DOI:
https://doi.org/10.1090/S0002-9947-1975-0358708-2

MathSciNet review:
0358708

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Abstract: The problem of embedding a topological space as a closed subspace of a product of members from a given family has received considerable attention in the past twenty years, while the corresponding problem in uniform spaces has been largely ignored. In this paper we initiate the study of the closed uniform subspaces of products of metric spaces. In §1 we introduce the functor , which is used in §2 to characterize the closed subspaces of products of metric spaces and separable metric spaces, and the closed subspaces of powers of the open unit interval . In §3 we obtain various descriptions of the functor which associates to each uniform space a closed subspace of a product of metric spaces and establish the equation . This leads to a characterzation of the completeness of , the uniform space generated by the countable -uniform covers, in terms of the completeness of and a countable intersection property on Cauchy filters.

**[C]**H. H. Corson,*The determination of paracompactness by uniformities*, Amer. J. Math.**80**(1958), 185-190. MR**20**#1292. MR**0094780 (20:1292)****[Fr]**Z. Frolik,*Interplay of measurable and uniform spaces*, Proc. Internat. Topology Conference in Yugoslavia, Budova, 1972. MR**0375243 (51:11439)****[Fr]**-,*A note on metric-fine spaces*, Proc. Amer. Math. Soc.**46**(1974), 111-119. MR**0358704 (50:11163)****[GI]**S. Ginsburg and J. R. Isbell,*Some operators on uniform spaces*, Trans. Amer. Math. Soc.**93**(1959), 145-168. MR**22**#2977. MR**0112119 (22:2977)****[GJ]**L. Gillman and M. Jerison,*Rings of continuous functions*, University Ser. in Higher Math., Van Nostrand, Princeton, N. J., 1960. MR**22**#6994. MR**0116199 (22:6994)****[H]**A. W. Hager,*Some nearly fine uniform spaces*, Proc. London Math. Soc. (3)**28**(1974), 517-546. MR**0397670 (53:1528)****[H]**-,*Perfect maps and epi-reflective hulls*, Canad. J. Math. (to appear). MR**0365499 (51:1751)****[HR]**A. W. Hager and M. D. Rice,*The commuting of coreflectors in uniform spaces with completion*, Czechoslovak. J. Math. (to appear). MR**0418051 (54:6095)****[HS]**M. Hušek,*The class of -compact spaces is simple*, Math Z.**110**(1969), 123-126. MR**39**#6260. MR**0244947 (39:6260)****[I]**J. R. Isbell,*Uniform spaces*, Math. Surveys, no. 12, Amer. Math. Soc., Providence, R. I., 1964. MR**30**#561. MR**0170323 (30:561)****[M]**K. Morita,*Topological completeness and -spaces*, Sci. Rep. Tokyo Kyoiku Daigaku Sect. A**10**(1970), 271-288. MR**42**#6785. MR**0271904 (42:6785)****[N]**S. A. Naimpally and B. D. Warrack,*Proximity spaces*, Cambridge Tracts in Math. and Math. Phys., no. 59, Cambridge Univ. Press, London and New York, 1970. MR**43**#3992. MR**0278261 (43:3992)****[NJ]**O. Njåstad,*On real-valued proximity mappings*, Math. Ann.**154**(1964), 413-419. MR**29**#4030. MR**0166757 (29:4030)****[R]**M. D. Rice,*Metric-fine uniform spaces*, J. London Math. Soc. (to appear).**[R]**-,*A short proof that metric spaces are realcompact*, Proc. Amer. Math. Soc.**32**(1972), 313-314. MR**0288724 (44:5920)****[R]**M. D. Rice,*Complete uniform spaces*, Proc. Second Internat. Topology Conference, Pittsburgh, Pa., 1972.**[RR]**M. D. Rice and G. D. Reynolds,*Covering properties of uniform spaces*, Oxford Quart. J. Math. (to appear).**[Re]**G. D. Reynolds,*Ultrafilters and epi-reflective hulls*(unpublished manuscript).**[S]**T. Shirota,*A class of topological spaces*, Osaka Math. J.**4**(1952), 23-40. MR**14**, 395. MR**0050872 (14:395b)****[Z]**P. Zenor,*Certain subsets of products of metacompact spaces and subparacompact spaces are realcompact*, Canad. J. Math.**24**(1972), 825-829. MR**46**#8181. MR**0309070 (46:8181)****[Z]**-,*Extensions of topological spaces*(to appear).

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DOI:
https://doi.org/10.1090/S0002-9947-1975-0358708-2

Article copyright:
© Copyright 1975
American Mathematical Society