Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-15T06:44:50.259Z Has data issue: false hasContentIssue false
  • English
  • Français

Homeomorphic Sets of Remote Points

Published online by Cambridge University Press:  20 November 2018

R. Grant Woods
R. Grant Woods*
Affiliation:
The University of Manitoba, Winnipeg, Manitoba
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let X be a completely regular Hausdorff space, and let βX denote the Stone-Čech compactification of X. A point p ∈ βX is called a remote point of βX if p does not belong to the βX-closure of any discrete subspace of X. Remote points were first defined and studied by Fine and Gillman, who proved that if the continuum hypothesis is assumed then the set of remote points of βR((βQ) is dense in βR – R(βQ – Q ) (R denotes the space of reals, Q the space of rationals). Assuming the continuum hypothesis, Plank has proved that if X is a locally compact, non-compact, separable metric space without isolated points, then βX has a set of remote points that is dense in βX – X. Robinson has extended this result by dropping the assumption that X is separable.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 3 , June 1971 , pp. 495 - 502
Copyright
Copyright © Canadian Mathematical Society 1971

References

1. Dugundji, J., Topology (Allyn and Bacon, Boston, 1965).Google Scholar
2. Fine, N. J. and Gillman, L., Extensions of continuous functions in (3Nt Bull. Amer. Math. Soc. 66 (1960), 376–381.Google Scholar
3. Fine, N. J. and Gillman, L., Remote points in βR, Proc. Amer. Math. Soc. 13 (1962), 29–36.Google Scholar
4. Gillman, L. and Jerison, M., Rings of continuous functions (Van Nostrand, New York, 1960).Google Scholar
5. Gleason, A. M., Projective topological spaces, Illinois J. Math. 2 (1958), 482–489.Google Scholar
6. Hocking, J. and Young, G., Topology (Addison-Wesley, Reading, 1961).Google Scholar
7. Isiwata, T., Mappings and spaces, Pacific J. Math. 20 (1967), 455–480.Google Scholar
8. Parovičenko, I. I., On a universal bicompactum of weight X (Russian), Dokl. Akad. Nauk B.S.S.R. 150 (1963), 36–39.Google Scholar
9. Plank, D. L., On a class of subalgebras of C(X) with applications to (3X-X, Fund. Math. 64 (1969), 41–54.Google Scholar
10. Robinson, S. M., Some properties of βX-X for complete spaces, Fund. Math. 64 (1969), 335–340.Google Scholar
11. Rudin, W., Homogeneity problems in the theory of Cech compactifications, Duke Math. J. 23 (1956), 409–419.Google Scholar
12. Sikorski, R., Boolean algebras (Springer, New York, second edition, 1964).Google Scholar
13. Woods, R. G., A Boolean algebra of regular closed subsets of (3X-X, Trans. Amer. Math. Soc. (to appear).Google Scholar