Almost all submaximal groups are paracompact and σ-discrete

O. Alas; I. Protasov; M. Tkačenko; V. Tkachuk; R. Wilson; I. Yaschenko

Fundamenta Mathematicae (1998)

  • Volume: 156, Issue: 3, page 241-260
  • ISSN: 0016-2736

Abstract

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We prove that any topological group of a non-measurable cardinality is hereditarily paracompact and strongly σ-discrete as soon as it is submaximal. Consequently, such a group is zero-dimensional. Examples of uncountable maximal separable spaces are constructed in ZFC.

How to cite

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Alas, O., et al. "Almost all submaximal groups are paracompact and σ-discrete." Fundamenta Mathematicae 156.3 (1998): 241-260. <http://eudml.org/doc/212271>.

@article{Alas1998,
abstract = {We prove that any topological group of a non-measurable cardinality is hereditarily paracompact and strongly σ-discrete as soon as it is submaximal. Consequently, such a group is zero-dimensional. Examples of uncountable maximal separable spaces are constructed in ZFC.},
author = {Alas, O., Protasov, I., Tkačenko, M., Tkachuk, V., Wilson, R., Yaschenko, I.},
journal = {Fundamenta Mathematicae},
keywords = {submaximality; maximality; paracompactness; topological group; separability; σ-discrete; strongly σ-discrete},
language = {eng},
number = {3},
pages = {241-260},
title = {Almost all submaximal groups are paracompact and σ-discrete},
url = {http://eudml.org/doc/212271},
volume = {156},
year = {1998},
}

TY - JOUR
AU - Alas, O.
AU - Protasov, I.
AU - Tkačenko, M.
AU - Tkachuk, V.
AU - Wilson, R.
AU - Yaschenko, I.
TI - Almost all submaximal groups are paracompact and σ-discrete
JO - Fundamenta Mathematicae
PY - 1998
VL - 156
IS - 3
SP - 241
EP - 260
AB - We prove that any topological group of a non-measurable cardinality is hereditarily paracompact and strongly σ-discrete as soon as it is submaximal. Consequently, such a group is zero-dimensional. Examples of uncountable maximal separable spaces are constructed in ZFC.
LA - eng
KW - submaximality; maximality; paracompactness; topological group; separability; σ-discrete; strongly σ-discrete
UR - http://eudml.org/doc/212271
ER -

References

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  1. [ArCo] A. V. Arhangel'skiĭ [A. V. Arkhangel'skiĭ] and P. J. Collins, z On submaximal spaces, Topology Appl. 64 (1995), 219-241. 
  2. [CoLF] W. W. Comfort and L. Feng, z The union of resolvable spaces is resolvable, Math. Japon. 38 (1993), 413-414. Zbl0769.54002
  3. [CoRo] W. W. Comfort and K. A. Ross, z Pseudocompactness and uniform continuity in topological groups, Pacific J. Math. 16 (1966), 483-496. Zbl0214.28502
  4. [vD] E. K. van Douwen, z Applications of minimal topologies, Topology Appl. 51 (1993), 125-139. Zbl0845.54028
  5. [En] R. Engelking, z General Topology, PWN, Warszawa, 1977. 
  6. [GRS] J. A. Guthrie, D. F. Reynolds and H. E. Stone, z Connected expansions of topologies, Bull. Austral. Math. Soc. 9 (1973), 259-265. Zbl0261.54002
  7. [Gu] I. Guran, z On topological groups close to being Lindelöf, Soviet Math. Dokl. 23 (1981), 173-175. Zbl0478.22002
  8. [He] E. Hewitt, z A problem of set-theoretic topology, Duke Math. J. 10 (1943), 309-333. 
  9. [HeRo] E. Hewitt and K. Ross, z Abstract Harmonic Analysis, Vol. 1, Springer, Berlin, 1963. 
  10. [Ma] V. I. Malykhin, z Extremally disconnected and similar groups, Soviet Math. Dokl. 16 (1975), 21-25. 
  11. [Pr] I. V. Protasov, z Absolutely resolvable groups, Ukrain. Math. Zh. 48 (1996), 383-392 (in Russian). 
  12. [Sch] J. Schröder, z On sub-, pseudo-, and quasimaximal spaces, in: Eighth Prague Topological Sympos., 1996, Abstracts, p. 22. 

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