Cartesian-closed coreflective subcategories of uniform spaces

Authors:
M. D. Rice and G. J. Tashjian

Journal:
Trans. Amer. Math. Soc. **276** (1983), 289-300

MSC:
Primary 54E15; Secondary 18B30, 18D15, 54B30

DOI:
https://doi.org/10.1090/S0002-9947-1983-0684509-6

MathSciNet review:
684509

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Abstract: This paper characterizes the coreflective subcategories of uniform spaces for which a natural function space structure generates the exponential law on . Such categories are cartesian-closed. Specifically, we show that is cartesian-closed in this way if and only if is inductively generated by a finitely productive family of locally fine spaces. The results divide naturally into two cases: those subcategories containing the unit interval are generated by precompact spaces, while the subcategories not containing the unit interval are generated by spaces which admit an infinite cardinal. These results may be used to derive the characterizations of cartesian-closed coreflective subcategories of Tychonoff spaces found in [**10**].

**[1]**Zdeněk Frolík,*Measurable uniform spaces*, Pacific J. Math.**55**(1974), 93–105. MR**383358****[2]**Seymour Ginsburg and J. R. Isbell,*Some operators on uniform spaces*, Trans. Amer. Math. Soc.**93**(1959), 145–168. MR**112119**, https://doi.org/10.1090/S0002-9947-1959-0112119-4**[3]**A. W. Hager,*Three classes of uniform spaces*, General topology and its relations to modern analysis and algebra, III (Proc. Third Prague Topological Sympos., 1971) Academia, Prague, 1972, pp. 159–164. MR**0353270****[4]**Horst Herrlich,*Cartesian closed topological categories*, Math. Colloq. Univ. Cape Town**9**(1974), 1–16. MR**460414****[5]**J. R. Isbell,*Uniform spaces*, Mathematical Surveys, No. 12, American Mathematical Society, Providence, R.I., 1964. MR**0170323****[6]**J. F. Kennison,*Reflective functors in general topology and elsewhere*, Trans. Amer. Math. Soc.**118**(1965), 303–315. MR**174611**, https://doi.org/10.1090/S0002-9947-1965-0174611-9**[7]**Saunders MacLane,*Categories for the working mathematician*, Springer-Verlag, New York-Berlin, 1971. Graduate Texts in Mathematics, Vol. 5. MR**0354798****[8]**L. D. Nel,*Cartesian closed coreflective hulls*, Quaestiones Math.**2**(1977/78), no. 1-3, 269–283. MR**480687****[9]**Gloria Tashjian,*Cartesian-closed coreflective subcategories of \underline{𝑈𝑛𝑖𝑓}*, General topology and its relations to modern analysis and algebra, IV (Proc. Fourth Prague Topological Sympos., Prague, 1976) Soc. Czechoslovak Mathematicians and Physicists, Prague, 1977, pp. 454–460. MR**0463260****[10]**Gloria Tashjian,*Cartesian-closed coreflective subcategories of Tychonoff spaces*, Pacific J. Math.**81**(1979), no. 2, 547–558. MR**547620****[11]**G. Tashjian and J. Vilímovský,*Coreflectors not preserving the interval and Baire partitions of uniform spaces*, Proc. Amer. Math. Soc.**77**(1979), no. 2, 257–263. MR**542094**, https://doi.org/10.1090/S0002-9939-1979-0542094-8

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1983-0684509-6

Article copyright:
© Copyright 1983
American Mathematical Society