| Title:
|
In quest of weaker connected topologies (English) |
| Author:
|
Tkačenko, M. G. |
| Author:
|
Tkachuk, V. V. |
| Author:
|
Uspenskii, V. V. |
| Author:
|
Wilson, R. G. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
37 |
| Issue:
|
4 |
| Year:
|
1996 |
| Pages:
|
825-841 |
| . |
| Category:
|
math |
| . |
| Summary:
|
We study when a topological space has a weaker connected topology. Various sufficient and necessary conditions are given for a space to have a weaker Hausdorff or regular connected topology. It is proved that the property of a space of having a weaker Tychonoff topology is preserved by any of the free topological group functors. Examples are given for non-preservation of this property by ``nice'' continuous mappings. The requirement that a space have a weaker Tychonoff connected topology is rather strong, but we show that it is difficult to construct spaces which would contain no infinite subspaces with a weaker connected $T_{3{1\over 2}}$-topology. (English) |
| Keyword:
|
connected |
| Keyword:
|
locally connected |
| Keyword:
|
free topological group |
| Keyword:
|
condensation |
| Keyword:
|
connectification |
| MSC:
|
22A05 |
| MSC:
|
54C10 |
| MSC:
|
54C25 |
| MSC:
|
54D05 |
| MSC:
|
54D06 |
| MSC:
|
54D10 |
| MSC:
|
54D25 |
| MSC:
|
54H11 |
| idZBL:
|
Zbl 0886.54034 |
| idMR:
|
MR1440714 |
| . |
| Date available:
|
2009-01-08T18:28:20Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/118891 |
| . |
| Reference:
|
[1] Alas O.T., Tkačenko M.G., Tkachuk V.V., Wilson R.G.: Connectifying some spaces.Topology and its Applications, to appear. MR 1397942 |
| Reference:
|
[2] Arhangel'skii A.V., Ponomarev V.I.: General Topology in Problems and Exercises (in Russian).Nauka Publishing House, Moscow, 1974. MR 0239550 |
| Reference:
|
[3] Arhangel'skii A.V.: The structure and classification of topological spaces and cardinal invariants (in Russian).Uspehi Mat. Nauk 33.6 (1978), 29-84. MR 0526012 |
| Reference:
|
[4] Engelking R.: General Topology.PWN, Warszawa, 1977. Zbl 0684.54001, MR 0500780 |
| Reference:
|
[5] Graev M.I.: Theory of topological groups, I (in Russian).Uspehi Mat. Nauk 5.2 (1950), 3-56. MR 0036245 |
| Reference:
|
[6] Miller A.W.: Special subsets of the real line.in: Handbook of Set-Theoretic Topology (Eds. K.Kunen, J.E.Vaughan), Elsevier S.P. B.V., 1984, pp.203-233. Zbl 0588.54035, MR 0776624 |
| Reference:
|
[7] Rudin M.E.: Martin's axiom.in: Handbook of Mathematical Logic, edited by J.Barwise, Amsterdam, North Holland P.C., 1977, pp.491-501. MR 0457132 |
| Reference:
|
[8] Shapirovsky B.E.: On mappings onto Tychonoff cubes (in Russian).Uspehi Mat. Nauk 35.3 (1980), 122-130. |
| Reference:
|
[9] Tkačenko M.G.: Examples of connected left separated spaces and topological groups.Acta Math., Acad. Sci. Hungar. 38.4 (1981), 257-261. MR 0647345 |
| Reference:
|
[10] Tkačenko M.G.: On group uniformities on the square of a space and extending pseudometrics.Bull. Austral. Math. Soc. 51.2 (1995), 309-335. MR 1322797 |
| Reference:
|
[11] Watson S., Wilson R.G.: Embeddings in connected spaces.Houston J. Math. 19.3 (1993), 469-481. Zbl 0837.54012, MR 1242433 |
| . |
