Locally -fine measurable spaces

Author:
Zdeněk Frolík

Journal:
Trans. Amer. Math. Soc. **196** (1974), 237-247

MSC:
Primary 54E15; Secondary 04A15, 28A05

DOI:
https://doi.org/10.1090/S0002-9947-1974-0383357-9

MathSciNet review:
0383357

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Abstract: Hyper-Baire sets and hyper-cozero sets in a uniform space are introduced, and it is shown that for metric-fine spaces the property ``every hypercozero set is a cozero set'' is equivalent to several much stronger properties like being locally *e*-fine (defined in §1), or having locally determined precompact part (introduced in §2). The metric-fine spaces with these additional properties form a coreflective subcategory of uniform spaces; the coreflection is explicitly described. The theory is applied to measurable uniform spaces. It is shown that measurable spaces with the additional properties mentioned above are coreflective and the coreflection is explicitly described. The two coreflections are not metrically determined.

**[E]**Čech,*Topological spaces*, 2nd ed., Publ. House Czech Acad. Sci., Prague, 1966; English transl., Wiley, New York, 1966. MR**35**#2254.**[Z]**Z. Frolík,*Topological methods in measure theory and the theory of measurable spaces*, General topology and its relations to modern analysis and algebra, III (Proc. Third Prague Topological Sympos., 1971) Academia, Prague, 1972, pp. 127–139. MR**0372141****1.**-, [F2]*Interplay af measurable and uniform spaces*, Topology and its Applications (Proc. 2nd Yugoslavia Sympos., Budva, 1972), Beograd, 1973, pp. 96-99.**2.**Zdeněk Frolík,*Hyper-extensions of 𝜎-algebras*, Comment. Math. Univ. Carolinae**14**(1973), 361–375. MR**0346115****3.**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**4.**Zdeněk Frolík,*Measurable uniform spaces*, Pacific J. Math.**55**(1974), 93–105. MR**0383358****[A]**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**5.**Anthony W. Hager,*Measurable uniform spaces*, Fund. Math.**77**(1972), no. 1, 51–73. MR**0324661****[R]**Hansell, [1] Thesis, Rochester, 1970.**6.**R. W. Hansell,*On the nonseparable theory of Borel and Souslin sets*, Bull. Amer. Math. Soc.**78**(1972), 236–241. MR**0294138**, https://doi.org/10.1090/S0002-9904-1972-12936-7**[J]**J. R. Isbell,*Uniform spaces*, Mathematical Surveys, No. 12, American Mathematical Society, Providence, R.I., 1964. MR**0170323**

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DOI:
https://doi.org/10.1090/S0002-9947-1974-0383357-9

Article copyright:
© Copyright 1974
American Mathematical Society