Cardinal functions on modifications of uniform spaces and fine uniform spaces
Author:
Věra Kurková
Journal:
Proc. Amer. Math. Soc. 84 (1982), 593600
MSC:
Primary 54E15; Secondary 54B30
MathSciNet review:
643756
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
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 pointcharacter (in dependence on ).
 [Č]
E. Čech, Topological spaces, Academia, Praha, 1966.
 [F]
Zdeněk
Frolík, Basic refinements of the category of uniform
spaces, TOPO 72—general topology and its applications (Proc.
Second Pittsburgh Internat. Conf., Pittsburgh, Pa., 1972; dedicated to the
memory of Johannes H. de Groot), Springer, Berlin, 1974,
pp. 140–158. Lecture Notes in Math., Vol. 378. MR 0358705
(50 #11164)
 [F]
Zdeněk
Frolík, Three technical tools in uniform spaces,
Seminar Uniform Spaces (Prague, 1973–1974) Mat. Ûstav
Československé Akad. Věd, Prague, 1975,
pp. 3–26. MR 0440510
(55 #13385)
 [I]
J.
R. Isbell, Uniform spaces, Mathematical Surveys, No. 12,
American Mathematical Society, 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), http://dx.doi.org/10.1090/S00029947196501746119
 [K]
Věra
KurkováPohlová, Fine and simply fine uniform
spaces, Seminar Uniform Spaces (Prague, 1973–1974) Mat.
Ûstav Československé Akad. Věd, Prague, 1975,
pp. 127–137. MR 0410680
(53 #14427)
 [K]
V.
KurkováPohlová, Fineness in the category of uniform
spaces, Topology, Vol. II (Proc. Fourth Colloq., Budapest, 1978)
Colloq. Math. Soc. János Bolyai, vol. 23, NorthHolland,
AmsterdamNew York, 1980, pp. 729–734. MR 588820
(81m:54050)
 [M]
Saunders
MacLane, Categories for the working mathematician,
SpringerVerlag, New YorkBerlin, 1971. Graduate Texts in Mathematics, Vol.
5. MR
0354798 (50 #7275)
 [P]
Jan
Pelant, Combinatorial properties of uniformities, General
topology and its relations to modern analysis and algebra, IV (Proc. Fourth
Prague Topological Sympos., Prague, 1976) Springer, Berlin, 1977,
pp. 154–165. Lecture Notes in Math., Vol. 609. MR 0500846
(58 #18360)
 [PR]
J. Pelant and V. Rödl, Coverings of infinite dimensional spaces (to appear).
 [R]
V. Rödl, Fineness in the category of all 0dimensional uniform spaces, Seminar Uniform Spaces 197374, ČSAV, Praha, 1975, pp. 139143.
 [R]
, Canonical partition relation and pointcharacter of spaces, Seminar Uniform Spaces 197677, pp. 7983.
 [R]
, Small spaces with a large pointcharacter (to appear).
 [V]
Jiří
Vilímovský, Categorical refinements and their
relation to reflective subcategories, Seminar Uniform Spaces (Prague,
1973–1974) Mat. Üstav Československé Akad. Ved,
Prague, 1975, pp. 83–111. MR 0413052
(54 #1173)
 [V]
, Reflections on distal spaces, Seminar Uniform Spaces 197576, ČSAV, Praha, 1976, pp. 6972.
 [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, SpringerVerlag, Berlin and New York, 1974, pp. 140158. MR 0358705 (50:11164)
 [F]
 , Three technical tools in uniform spaces, Seminar Uniform Spaces, 197374, ČSAV, Praha, 1975, pp. 326. 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), 303315. MR 0174611 (30:4812)
 [K]
 V. Kůrková, Fine and simply fine uniform spaces, Seminar Uniform Spaces 197374, ČSAV, Praha, 1975, pp. 127137. MR 0410680 (53:14427)
 [K]
 , Fineness in the category of uniform spaces, Proc. Colloq. on Topology, Budapest, 1978, pp. 729734. MR 588820 (81m:54050)
 [M]
 S. Mac Lane, Categories for the working mathematician, SpringerVerlag, 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. 154165. MR 0500846 (58:18360)
 [PR]
 J. Pelant and V. Rödl, Coverings of infinite dimensional spaces (to appear).
 [R]
 V. Rödl, Fineness in the category of all 0dimensional uniform spaces, Seminar Uniform Spaces 197374, ČSAV, Praha, 1975, pp. 139143.
 [R]
 , Canonical partition relation and pointcharacter of spaces, Seminar Uniform Spaces 197677, pp. 7983.
 [R]
 , Small spaces with a large pointcharacter (to appear).
 [V]
 J. Vilímovský, Categorial refinements and their relation to reflective subcategories, Seminar Uniform Spaces 197374, ČSAV, Praha, 1975, pp. 83111. MR 0413052 (54:1173)
 [V]
 , Reflections on distal spaces, Seminar Uniform Spaces 197576, ČSAV, Praha, 1976, pp. 6972.
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
DOI:
http://dx.doi.org/10.1090/S00029939198206437564
PII:
S 00029939(1982)06437564
Keywords:
Uniform space,
modification,
cardinal modification,
pointcharacter
Article copyright:
© Copyright 1982
American Mathematical Society
