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)



Filter characterizations of $ C$- and $ C\sp{\ast} $-embeddings

Author: John William Green
Journal: Proc. Amer. Math. Soc. 35 (1972), 574-580
MSC: Primary 54C45
MathSciNet review: 0303497
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A filter F on a space S is completely regular if the complement of each set in F is completely separated from some set in F. A characterization of the Stone-Čech compactification due to Alexandroff is used to establish the following theorem. Suppose K is a subspace of a Tychonoff space S. K is $ {C^ \ast }$-embedded in S if and only if the trace on K of every maximal completely regular filter on S intersecting K is maximal completely regular on K. A similar characterization of the C-embedded subsets of a Tychonoff space is obtained as are several related results.

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

  • [1] P. S. Aleksandrov, Bikompakte Erweiterung topologischer Räume, Mat. Sb. 5 (47) (1939), 403-423. (Russian) MR 1, 318.
  • [2] R. A. Alo and H. L. Shapiro, $ \mathcal{L}$-realcompactifications and normal bases, J. Austral. Math. Soc. 9 (1969), 489-495. MR 39 #3455. MR 0242121 (39:3455)
  • [3] B. Banaschewski, Über zwei Extremaleigenschaften topologischer Räume, Math. Nachr. 13 (1955), 141-150. MR 17, 66. MR 0070993 (17:66h)
  • [4] N. Bourbaki, Éléments de mathématique. Part. 1. Les structures fondamentales de l'analyse. Livre III: Topologie générale, Actualités Sci. Indust., no. 1029, Hermann, Paris, 1947; English transl., Addison-Wesley, Reading, Mass., 1966. MR 9, 261; MR 34 #5044b.
  • [5] Ky Fan and N. Gottesman, On compactifications of Freudenthal and Wallman, Nederl. Akad. Wetensch. Proc. Ser. A 55=Indag. Math. 14 (1952), 504-510. MR 14, 669. MR 0052761 (14:669c)
  • [6] L. Gillman and M. Jerison, Rings of continuous functions, University Series in Higher Math., Van Nostrand, Princeton, N.J., 1960. MR 22 #6994. MR 0116199 (22:6994)
  • [7] O. Njästad, A note on compactification by bounding systems, J. London Math. Soc. 40 (1965), 526-532. MR 33 #1832. MR 0193614 (33:1832)
  • [8] R. M. Stephenson, Jr., Product spaces for which the Stone-Weierstrass theorem holds, Proc. Amer. Math. Soc. 21 (1969), 284-288. MR 40 #3499. MR 0250260 (40:3499)
  • [9] F. J. Wagner, Notes on compactification. I, II, Nederl. Akad. Wetensch. Proc. Ser. A 60=Indag. Math. 19 (1957), 171-176, 177-181. MR 19, 436. MR 0087919 (19:436c)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 54C45

Retrieve articles in all journals with MSC: 54C45

Additional Information

Keywords: Stone-Čech compactification, z-filter, completely regular filter, $ {C^ \ast }$-embedding, C-embedding
Article copyright: © Copyright 1972 American Mathematical Society

American Mathematical Society