Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-24T09:37:59.040Z Has data issue: false hasContentIssue false
  • English
  • Français

Nowhere Dense Subsets of Metric Spaces with Applications to Stone-Cech Compactifications

Published online by Cambridge University Press:  20 November 2018

Jack R. Porter  and
R. Grant Woods
Jack R. Porter
Affiliation:
University of Kansas, Lawrence, Kansas
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 metric space. Assume either that X is locally compact or that X has no more than countably many isolated points. It is proved that if F is a nowhere dense subset of X, then it is regularly nowhere dense (in the sense of Katětov) and hence is contained in the topological boundary of some regular-closed subset of X. This result is used to obtain new properties of the remote points of the Stone-Čech compactification of a metric space without isolated points.

Let βX denote the Stone-Čech compactification of the completely regular Hausdorff space X. Fine and Gillman [3] define a point p of βX to be remote if p is not in the βX-closure of a discrete subset of X.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 24 , Issue 4 , August 1972 , pp. 622 - 630
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Dugundji, J., Topology (Allyn and Bacon, Mass., 1965).Google Scholar
2. Fine, N. J. and Gillman, L., Extensions of continuous functions in f3N, Bull. Amer. Math. Soc. 66 (1960), 376-381.Google Scholar
3. Fine, N. J. and Gillman, L., Remote points in 0R, Proc. Amer. Math. Soc. 13 (1962), 29–36.Google Scholar
4. Gillman, L. and Jerison, M., Rings of continuous functions, University Series in Higher Mathematics (Van Nostrand, Princeton, New Jersey, 1960).Google Scholar
5. Katětov, M., On the equivalence of certain types of extensions of topological spaces, Časopis Pěst. Mat. Fys. 72 (1947), 101–106.Google Scholar
6. Mandelker, M., Prime z-ideal structure of C(R), Fund. Math. 63 (1968), 145–166.Google Scholar
7. Mandelker, M., Round z-filters and round subsets of (3X, Israel J. Math. 7 (1969), 1–8.Google Scholar
8. Plank, D. L., On a class of subalgebras of C﹛X) with applications to βX-X, Fund. Math. 64 (1969), 41–52.Google Scholar
9. Sikorski, R., Boolean algebras, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, Neue Folge, Band 25 (Academic Press, New York; Springer-Verlag, Berlin and New York, 1964).Google Scholar
10. Woods, R. G., Co-absolutes of remainders of Stone-Čech compactifications, Pacific J. Math. 37 (1971), 545–560.Google Scholar