In quest of weaker connected topologies
Tkachenko, Mihail G. ; Tkachuk, Vladimir Vladimirovich ; Uspenskij, Vladimir Vladimirovich ; Wilson, Richard Gordon
Commentationes Mathematicae Universitatis Carolinae, Tome 37 (1996), p. 825-841 / Harvested from Czech Digital Mathematics Library
Access to full text
Full (PDF)

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.

Publié le : 1996-01-01
Classification:  22A05,  54C10,  54C25,  54D05,  54D06,  54D10,  54D25,  54H11 
@article{118891,
     author = {Mihail G. Tkachenko and Vladimir Vladimirovich Tkachuk and Vladimir Vladimirovich Uspenskij and Richard Gordon Wilson},
     title = {In quest of weaker connected topologies},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     volume = {37},
     year = {1996},
     pages = {825-841},
     zbl = {0886.54034},
     mrnumber = {1440714},
     language = {en},
     url = {http://dml.mathdoc.fr/item/118891}
}

Tkachenko, Mihail G.; Tkachuk, Vladimir Vladimirovich; Uspenskij, Vladimir Vladimirovich; Wilson, Richard Gordon. In quest of weaker connected topologies. Commentationes Mathematicae Universitatis Carolinae, Tome 37 (1996) pp. 825-841. http://gdmltest.u-ga.fr/item/118891/

Alas O.T.; Tkačenko M.G.; Tkachuk V.V.; Wilson R.G. Connectifying some spaces, Topology and its Applications, to appear. | MR 1397942

Arhangel'Skii A.V.; Ponomarev V.I. General Topology in Problems and Exercises (in Russian), Nauka Publishing House, Moscow, 1974. | MR 0239550

Arhangel'Skii A.V. The structure and classification of topological spaces and cardinal invariants (in Russian), Uspehi Mat. Nauk 33.6 (1978), 29-84. (1978) | MR 0526012

Engelking R. General Topology, PWN, Warszawa, 1977. | MR 0500780 | Zbl 0684.54001

Graev M.I. Theory of topological groups, I (in Russian), Uspehi Mat. Nauk 5.2 (1950), 3-56. (1950) | MR 0036245

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. | MR 0776624 | Zbl 0588.54035

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

Shapirovsky B.E. On mappings onto Tychonoff cubes (in Russian), Uspehi Mat. Nauk 35.3 (1980), 122-130. (1980)

Tkačenko M.G. Examples of connected left separated spaces and topological groups, Acta Math., Acad. Sci. Hungar. 38.4 (1981), 257-261. (1981) | MR 0647345

Tkačenko M.G. On group uniformities on the square of a space and extending pseudometrics, Bull. Austral. Math. Soc. 51.2 (1995), 309-335. (1995) | MR 1322797

Watson S.; Wilson R.G. Embeddings in connected spaces, Houston J. Math. 19.3 (1993), 469-481. (1993) | MR 1242433 | Zbl 0837.54012