Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-26T00:37:43.776Z Has data issue: false hasContentIssue false
  • English
  • Français

Hyperspaces of H-Closed Spaces

Published online by Cambridge University Press:  20 November 2018

L. M. Friedler ,
R. F. Dickman Jr.  and
R. L. Krystock
L. M. Friedler
Affiliation:
College of St. Scholastica, Duluth, Minnesota
R. F. Dickman Jr.
Affiliation:
Virginia Polytechnic Institute, Blacksburg, Virginia
R. L. Krystock
Affiliation:
Virginia Polytechnic Institute, Blacksburg, Virginia
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.

A space is H(i) [R(i)] if every open [regular] filter base has a cluster point and H(ii) [R(ii)] if every open [regular] filter base with a unique cluster point converges. This terminology is due to C. T. Scarborough and A. H. Stone [11]; H(i) spaces have been studied as quasi-H-closed spaces in [10] and as generalized absolutely closed spaces in [6]. Hausdorff H(i) [H(ii)] spaces are called H-closed [minimal Hausdorff] and regular T1 R(i) [R(ii)] spaces are called R-closed [minimal regular]. For a space X, 2X is the set of all non-empty closed subsets of X with the finite topology [8]. The present study was motivated by the longstanding problem of whether or not a T3 space with every closed subset R-closed is compact, and also by the well-known result ([8] and [14]) that X is compact if and only if 2X is compact.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 5 , 01 October 1980 , pp. 1072 - 1079
Copyright
Copyright © Canadian Mathematical Society 1980

References

1. Berri, M. P., Porter, J. R. and Stephenson, R. M. Jr., A survey of minimal topological spaces,in General topology and its relations to modern analysis and algebra III, Proc. Kanpur Top. Conf. (1903), Academic Press, N.Y. (1970).Google Scholar
2. Dickman, R. F. Jr. and Porter, J. R., θ-closed subsets of Hausdorff spaces, Pacific J. Math. 50 (1975), 407–415.Google Scholar
3. Ginsburg, J., Some results on the countable compactness and pseudocompactness of hyperspaces, Can. J. Math. 27 (1975), 1392–1399.Google Scholar
4. Halmos, P. R., Lectures on Boolean algebras (Springer-Yerlag, New York, 1974).CrossRefGoogle Scholar
5. Keesling, J., Normality and properties related to compactness in hyperspaces, Proc. Amer. Math. Soc. 24 (1970), 760–706.Google Scholar
6. Liu, C. T., Absolutely closed spaces, Trans. Amer. Math. Soc. 130 (1968), 80–104.Google Scholar
7. Kuratowski, K., Topology, Volumes I, II, Academic Press (1960), 160–180 (Vol. I), 45–53 (Vol. II).Google Scholar
8. Michael, E., Topologies on spaces of subsets, Trans. Amer. Math. Soc 71 (1951), 152–182.Google Scholar
9. Pettey, D. H., Products of R-closed spaces, Notices Amer. Math. Soc. 24 (1977).Google Scholar
10. Porter, J. and Thomas, J., On H-closed spaces and minimal Hausdorff spaces, Trans. Amer. Math. Soc. 138 (1969), 159–170.Google Scholar
11. Scarborough, C. T. and Stone, A. H., Products of nearly compact spaces, Trans. Amer. Math. Soc. 124 (1966), 131–147.Google Scholar
12. Stephenson, R. M. Jr., Pseudocompact spaces, Trans. Amer. Math. Soc. 134 (1968), 437–448.Google Scholar
13. Velicko, N. V., H-closed topological spaces, Mat. Sb. 70 (112), (1966), 98–112.Google Scholar
14. Vietoris, P., Bereiche Zewiter Ordnung, Monatshefte fur Matematik und Physik 33 (1923), 49–62.Google Scholar
15. Viglino, G., Semi-normal and c-compact spaces, Duke Math. J. 38 (1971), 57–01.Google Scholar
16. Willard, S., General topology (Addison-Wesley, Reading, Mass., 1970).Google Scholar