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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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**0094780**, https://doi.org/10.2307/2372828**[Fr]**Zdeněk Frolík,*Interplay of measurable and uniform spaces*, Proceedings of the International Symposium on Topology and its Applications (Budva, 1972) Savez Društava Mat. Fiz. i Astronom., Belgrade, 1973, pp. 98–101. MR**0375243****[Fr]**Zdeněk Frolík,*A note on metric-fine spaces*, Proc. Amer. Math. Soc.**46**(1974), 111–119. MR**0358704**, https://doi.org/10.1090/S0002-9939-1974-0358704-X**[GI]**Seymour Ginsburg and J. R. Isbell,*Some operators on uniform spaces*, Trans. Amer. Math. Soc.**93**(1959), 145–168. MR**0112119**, https://doi.org/10.1090/S0002-9947-1959-0112119-4**[GJ]**Leonard Gillman and Meyer Jerison,*Rings of continuous functions*, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR**0116199****[H]**Anthony W. Hager,*Some nearly fine uniform spaces*, Proc. London Math. Soc. (3)**28**(1974), 517–546. MR**0397670**, https://doi.org/10.1112/plms/s3-28.3.517**[H]**Anthony W. Hager,*Perfect maps and epi-reflective hulls*, Canad. J. Math.**27**(1975), 11–24. MR**0365499**, https://doi.org/10.4153/CJM-1975-003-6**[HR]**Anthony W. Hager and Michael D. Rice,*The commuting of coreflectors in uniform spaces with completion*, Czechoslovak Math. J.**26(101)**(1976), no. 3, 371–380. MR**0418051****[HS]**Miroslav Hušek,*The class of 𝑘-compact spaces is simple*, Math. Z.**110**(1969), 123–126. MR**0244947**, https://doi.org/10.1007/BF01124977**[I]**J. R. Isbell,*Uniform spaces*, Mathematical Surveys, No. 12, American Mathematical Society, Providence, R.I., 1964. MR**0170323****[M]**Kiiti Morita,*Topological completions and 𝑀-spaces*, Sci. Rep. Tokyo Kyoiku Daigaku Sect. A**10**(1970), 271–288 (1970). MR**0271904****[N]**S. A. Naimpally and B. D. Warrack,*Proximity spaces*, Cambridge Tracts in Mathematics and Mathematical Physics, No. 59, Cambridge University Press, London-New York, 1970. MR**0278261****[NJ]**Olav NjÈ§stad,*On real-valued proximity mappings*, Math. Ann.**154**(1964), 413–419. MR**0166757**, https://doi.org/10.1007/BF01375524**[R]**M. D. Rice,*Metric-fine uniform spaces*, J. London Math. Soc. (to appear).**[R]**M. D. Rice,*A short proof that metric spaces are realcompact*, Proc. Amer. Math. Soc.**32**(1972), 313–314. MR**0288724**, https://doi.org/10.1090/S0002-9939-1972-0288724-3**[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]**Taira Shirota,*A class of topological spaces*, Osaka Math. J.**4**(1952), 23–40. MR**0050872****[Z]**Phillip Zenor,*Certain subsets of products of metacompact spaces and subparacompact spaces are realcompact*, Canad. J. Math.**24**(1972), 825–829. MR**0309070**, https://doi.org/10.4153/CJM-1972-081-9**[Z]**-,*Extensions of topological spaces*(to appear).

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
54E15

Retrieve articles in all journals with MSC: 54E15

Additional Information

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

Article copyright:
© Copyright 1975
American Mathematical Society