| Title:
|
Normal Vietoris implies compactness: a short proof (English) |
| Author:
|
Maio, G. Di |
| Author:
|
Meccariello, E. |
| Author:
|
Naimpally, S. |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
54 |
| Issue:
|
1 |
| Year:
|
2004 |
| Pages:
|
181-182 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
One of the most celebrated results in the theory of hyperspaces says that if the Vietoris topology on the family of all nonempty closed subsets of a given space is normal, then the space is compact (Ivanova-Keesling-Velichko). The known proofs use cardinality arguments and are long. In this paper we present a short proof using known results concerning Hausdorff uniformities. (English) |
| Keyword:
|
hyperspaces |
| Keyword:
|
Vietoris topology |
| Keyword:
|
locally finite topology |
| Keyword:
|
Hausdorff metric |
| Keyword:
|
compactness |
| Keyword:
|
normality |
| Keyword:
|
countable compactness |
| MSC:
|
54B20 |
| MSC:
|
54D30 |
| MSC:
|
54E15 |
| idZBL:
|
Zbl 1049.54010 |
| idMR:
|
MR2040229 |
| . |
| Date available:
|
2009-09-24T11:10:59Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127874 |
| . |
| Reference:
|
[1] G. Beer: Topologies on Closed and Closed Convex Sets.Kluwer Academic Publishers, , 1993. Zbl 0792.54008, MR 1269778 |
| Reference:
|
[2] A. Di Concilio, S. A. Naimpally and P. L. Sharma: Proximal hypertopologies.Proceedings of the VI Brasilian Topological Meeting, Campinas, Brazil (1988), Unpublished. |
| Reference:
|
[3] R. Engelking: General Topology.Helderman Verlag, Berlin, 1989, Revised and completed version. Zbl 0684.54001, MR 1039321 |
| Reference:
|
[4] V. M. Ivanova: On the theory of the space of subsets.Dokl. Akad. Nauk. SSSR 101 (1955), 601–603. MR 0069479 |
| Reference:
|
[5] J. Keesling: Normality and properties related to compactness in hyperspaces.Proc. Amer. Math. Soc. 24 (1970), 760–766. Zbl 0189.53203, MR 0253292, 10.1090/S0002-9939-1970-0253292-7 |
| Reference:
|
[6] J. Keesling: On the equivalence of normality and compactness in hyperspaces.Pacific J. Math. 33 (1970), 657–667. Zbl 0182.25401, MR 0267516, 10.2140/pjm.1970.33.657 |
| Reference:
|
[7] S. A. Naimpally and P. L. Sharma: Fine uniformity and the locally finite hyperspace topology on $2^X$.Proc. Amer. Math. Soc. 103 (1988), 641–646. MR 0943098, 10.1090/S0002-9939-1988-0943098-9 |
| Reference:
|
[8] N. V. Velichko: On spaces of closed subsets.Sibirskii Matem. Z. 16 (1975), 627–629. |
| . |
