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)

 
 

 

Properties of Stone-Čech compactifications of discrete spaces


Author: Nancy M. Warren
Journal: Proc. Amer. Math. Soc. 33 (1972), 599-606
MSC: Primary 54D35
DOI: https://doi.org/10.1090/S0002-9939-1972-0292035-X
MathSciNet review: 0292035
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \beta N$ be the Stone-Čech compactification of the integers N. It is shown that p is a P-point of $ \beta N - N$, then $ \beta N - N - \{ p\} $ is not normal. Let D be an uncountable discrete set and $ {E_0}$ be the set of points in $ \beta D - D$ in the closures of countable subsets of D It is shown that there is a two-valued continuous function on $ {E_0}$ which cannot be extended continuously to $ \beta D$.


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

  • [1] W. W. Comfort and S. Negrepontis, Homeomorphs of three subspaces of $ \beta N/N$, Math. Z. 107 (1968), 53-58. MR 38 #2739. MR 0234422 (38:2739)
  • [2] N. J. Fine and L. Gillman, Extension of continuous functions in $ \beta N$, Bull. Amer. Math. Soc. 66 (1960), 376-381. MR 23 #A619. MR 0123291 (23:A619)
  • [3] M. E. Rudin, Types of ultrafilters, Topology Seminar (Wisconsin, 1965), Ann. of Math. Studies, no. 60, Princeton Univ. Press, Princeton, N.J., 1966, p. 148. MR 35 #7284. MR 0216451 (35:7284)
  • [4] G. Kurepa, Ensembles linéaires et une classe de tableaux ramifiés (tableaux ramifiés de M. Aronszajn), Publ. Math. Univ. Belgrade 6 (1937), 129-160.
  • [5] F. B. Jones, On certain well-ordered monotone collections of sets, J. Elisha Mitchell Sci. Soc. 69 (1953), 33-34. MR 15, 18; 1139. MR 0056052 (15:18b)
  • [6] L. Gillman and M. Jerison, Rings of continuous functions, The University Series in Higher Math., Van Nostrand, Princeton, N.J., 1960, p. 86. MR 22 #6994. MR 0116199 (22:6994)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 54D35

Retrieve articles in all journals with MSC: 54D35


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0292035-X
Keywords: Stone-Čech compactification, P-point, $ {C^\ast}$-embedding, discrete spaces
Article copyright: © Copyright 1972 American Mathematical Society

American Mathematical Society