Cardinal functions on modifications of uniform spaces and fine uniform spaces

Author:
Věra Kurková

Journal:
Proc. Amer. Math. Soc. **84** (1982), 593-600

MSC:
Primary 54E15; Secondary 54B30

DOI:
https://doi.org/10.1090/S0002-9939-1982-0643756-4

MathSciNet review:
643756

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Abstract: The paper studies the question for which modifications of Unif the following theorem can be generalized by substituting a precompact modification by : A uniform space has the finest uniformity inducing its proximity if and only if each proximally continuous mapping from this space to any other uniform space is uniformly continuous. By means of two cardinal functions defined on the class of all modifications of Unif there is shown that this is possible only for cardinal modifications . Assuming GCH, the problem for cardinal modifications is solved for uniform spaces of a limited point-character (in dependence on ).

**[Č]**E. Čech,*Topological spaces*, Academia, Praha, 1966.**[F]**Z. Frolík,*Basic refinements of uniform spaces*, General Topology and its Applications (R. A. Alo, R. W. Heath and J. Nagota, Eds.), Lecture Notes in Math., no. 378, Springer-Verlag, Berlin and New York, 1974, pp. 140-158. MR**0358705 (50:11164)****[F]**-,*Three technical tools in uniform spaces*, Seminar Uniform Spaces, 1973-74, ČSAV, Praha, 1975, pp. 3-26. MR**0440510 (55:13385)****[I]**J. R. Isbell,*Uniform spaces*, Math. Surveys, no. 12, Amer. Math. Soc., Providence, R. I., 1964. MR**0170323 (30:561)****[K]**J. F. Kennison,*Reflective functors in general topology and elsewhere*, Trans. Amer. Math. Soc.**118**(1965), 303-315. MR**0174611 (30:4812)****[K]**V. Kůrková,*Fine and simply fine uniform spaces*, Seminar Uniform Spaces 1973-74, ČSAV, Praha, 1975, pp. 127-137. MR**0410680 (53:14427)****[K]**-,*Fineness in the category of uniform spaces*, Proc. Colloq. on Topology, Budapest, 1978, pp. 729-734. MR**588820 (81m:54050)****[M]**S. Mac Lane,*Categories for the working mathematician*, Springer-Verlag, New York, 1971. MR**0354798 (50:7275)****[P]**J. Pelant,*Combinatorial properties of uniformities*, General Topology and its Relation to Modern Analysis and Algebra IV, Prague, 1976, pp. 154-165. MR**0500846 (58:18360)****[P-R]**J. Pelant and V. Rödl,*Coverings of infinite dimensional spaces*(to appear).**[R]**V. Rödl,*Fineness in the category of all*0*-dimensional uniform spaces*, Seminar Uniform Spaces 1973-74, ČSAV, Praha, 1975, pp. 139-143.**[R]**-,*Canonical partition relation and point-character of**-spaces*, Seminar Uniform Spaces 1976-77, pp. 79-83.**[R]**-,*Small spaces with a large point-character*(to appear).**[V]**J. Vilímovský,*Categorial refinements and their relation to reflective subcategories*, Seminar Uniform Spaces 1973-74, ČSAV, Praha, 1975, pp. 83-111. MR**0413052 (54:1173)****[V]**-,*Reflections on distal spaces*, Seminar Uniform Spaces 1975-76, ČSAV, Praha, 1976, pp. 69-72.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1982-0643756-4

Keywords:
Uniform space,
modification,
cardinal modification,
point-character

Article copyright:
© Copyright 1982
American Mathematical Society