Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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
MathSciNet review: 643756
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The paper studies the question for which modifications $ r$ of Unif the following theorem can be generalized by substituting a precompact modification $ p$ by $ r$: 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 $ {p^\alpha }$. Assuming GCH, the problem for cardinal modifications $ {p^\alpha }$ is solved for uniform spaces of a limited point-character (in dependence on $ \alpha $).

References [Enhancements On Off] (What's this?)

  • [Č] E. Čech, Topological spaces, Academia, Praha, 1966.
  • [F$ _{1}$] Z. Frolík, Basic refinements of uniform spaces, General Topology and its Applications (R. A. Alo, R. W. Heath and J. Nagota, Eds.), Lecture Notes in Math., no. 378, Springer-Verlag, Berlin and New York, 1974, pp. 140-158. MR 0358705 (50:11164)
  • [F$ _{2}$] -, Three technical tools in uniform spaces, Seminar Uniform Spaces, 1973-74, ČSAV, Praha, 1975, pp. 3-26. MR 0440510 (55:13385)
  • [I] J. R. Isbell, Uniform spaces, Math. Surveys, no. 12, Amer. Math. Soc., Providence, R. I., 1964. MR 0170323 (30:561)
  • [K] J. F. Kennison, Reflective functors in general topology and elsewhere, Trans. Amer. Math. Soc. 118 (1965), 303-315. MR 0174611 (30:4812)
  • [K$ _{1}$] V. Kůrková, Fine and simply fine uniform spaces, Seminar Uniform Spaces 1973-74, ČSAV, Praha, 1975, pp. 127-137. MR 0410680 (53:14427)
  • [K$ _{2}$] -, Fineness in the category of uniform spaces, Proc. Colloq. on Topology, Budapest, 1978, pp. 729-734. MR 588820 (81m:54050)
  • [M] S. Mac Lane, Categories for the working mathematician, Springer-Verlag, New York, 1971. MR 0354798 (50:7275)
  • [P] J. Pelant, Combinatorial properties of uniformities, General Topology and its Relation to Modern Analysis and Algebra IV, Prague, 1976, pp. 154-165. MR 0500846 (58:18360)
  • [P-R] J. Pelant and V. Rödl, Coverings of infinite dimensional spaces (to appear).
  • [R$ _{1}$] V. Rödl, Fineness in the category of all 0-dimensional uniform spaces, Seminar Uniform Spaces 1973-74, ČSAV, Praha, 1975, pp. 139-143.
  • [R$ _{2}$] -, Canonical partition relation and point-character of $ {l_1}$-spaces, Seminar Uniform Spaces 1976-77, pp. 79-83.
  • [R$ _{3}$] -, Small spaces with a large point-character (to appear).
  • [V$ _{1}$] J. Vilímovský, Categorial refinements and their relation to reflective subcategories, Seminar Uniform Spaces 1973-74, ČSAV, Praha, 1975, pp. 83-111. MR 0413052 (54:1173)
  • [V$ _{2}$] -, Reflections on distal spaces, Seminar Uniform Spaces 1975-76, ČSAV, Praha, 1976, pp. 69-72.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 54E15, 54B30

Retrieve articles in all journals with MSC: 54E15, 54B30

Additional Information

Keywords: Uniform space, modification, cardinal modification, point-character
Article copyright: © Copyright 1982 American Mathematical Society

American Mathematical Society