In quest of weaker connected topologies

Mihail G. Tkachenko; Vladimir Vladimirovich Tkachuk; Vladimir Vladimirovich Uspenskij; Richard Gordon Wilson

Commentationes Mathematicae Universitatis Carolinae (1996)

  • Volume: 37, Issue: 4, page 825-841
  • ISSN: 0010-2628

Abstract

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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 2 -topology.

How to cite

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Tkachenko, Mihail G., et al. "In quest of weaker connected topologies." Commentationes Mathematicae Universitatis Carolinae 37.4 (1996): 825-841. <http://eudml.org/doc/247940>.

@article{Tkachenko1996,
abstract = {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.},
author = {Tkachenko, Mihail G., Tkachuk, Vladimir Vladimirovich, Uspenskij, Vladimir Vladimirovich, Wilson, Richard Gordon},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {connected; locally connected; free topological group; condensation; connectification; connected; locally connected; free topological group; condensation; connectification},
language = {eng},
number = {4},
pages = {825-841},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {In quest of weaker connected topologies},
url = {http://eudml.org/doc/247940},
volume = {37},
year = {1996},
}

TY - JOUR
AU - Tkachenko, Mihail G.
AU - Tkachuk, Vladimir Vladimirovich
AU - Uspenskij, Vladimir Vladimirovich
AU - Wilson, Richard Gordon
TI - In quest of weaker connected topologies
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1996
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 37
IS - 4
SP - 825
EP - 841
AB - 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.
LA - eng
KW - connected; locally connected; free topological group; condensation; connectification; connected; locally connected; free topological group; condensation; connectification
UR - http://eudml.org/doc/247940
ER -

References

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  1. Alas O.T., Tkačenko M.G., Tkachuk V.V., Wilson R.G., Connectifying some spaces, Topology and its Applications, to appear. MR1397942
  2. Arhangel'skii A.V., Ponomarev V.I., General Topology in Problems and Exercises (in Russian), Nauka Publishing House, Moscow, 1974. MR0239550
  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. (1978) MR0526012
  4. Engelking R., General Topology, PWN, Warszawa, 1977. Zbl0684.54001MR0500780
  5. Graev M.I., Theory of topological groups, I (in Russian), Uspehi Mat. Nauk 5.2 (1950), 3-56. (1950) MR0036245
  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. Zbl0588.54035MR0776624
  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. MR0457132
  8. Shapirovsky B.E., On mappings onto Tychonoff cubes (in Russian), Uspehi Mat. Nauk 35.3 (1980), 122-130. (1980) 
  9. Tkačenko M.G., Examples of connected left separated spaces and topological groups, Acta Math., Acad. Sci. Hungar. 38.4 (1981), 257-261. (1981) MR0647345
  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. (1995) MR1322797
  11. Watson S., Wilson R.G., Embeddings in connected spaces, Houston J. Math. 19.3 (1993), 469-481. (1993) Zbl0837.54012MR1242433

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