On topological and algebraic structure of extremally disconnected semitopological groups

Aleksander V. Arhangel'skii

Commentationes Mathematicae Universitatis Carolinae (2000)

  • Volume: 41, Issue: 4, page 803-810
  • ISSN: 0010-2628

Abstract

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Starting with a very simple proof of Frol’ık’s theorem on homeomorphisms of extremally disconnected spaces, we show how this theorem implies a well known result of Malychin: that every extremally disconnected topological group contains an open and closed subgroup, consisting of elements of order 2 . We also apply Frol’ık’s theorem to obtain some further theorems on the structure of extremally disconnected topological groups and of semitopological groups with continuous inverse. In particular, every Lindelöf extremally disconnected semitopological group with continuous inverse and with square roots is countable, and every extremally disconnected topological field is discrete.

How to cite

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Arhangel'skii, Aleksander V.. "On topological and algebraic structure of extremally disconnected semitopological groups." Commentationes Mathematicae Universitatis Carolinae 41.4 (2000): 803-810. <http://eudml.org/doc/248631>.

@article{Arhangelskii2000,
abstract = {Starting with a very simple proof of Frol’ık’s theorem on homeomorphisms of extremally disconnected spaces, we show how this theorem implies a well known result of Malychin: that every extremally disconnected topological group contains an open and closed subgroup, consisting of elements of order $2$. We also apply Frol’ık’s theorem to obtain some further theorems on the structure of extremally disconnected topological groups and of semitopological groups with continuous inverse. In particular, every Lindelöf extremally disconnected semitopological group with continuous inverse and with square roots is countable, and every extremally disconnected topological field is discrete.},
author = {Arhangel'skii, Aleksander V.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {extremally disconnected; semitopological group; order 2; Souslin number; Lindelöf space; extremally disconnected; semitopological group; order 2; Souslin number; Lindelöf space},
language = {eng},
number = {4},
pages = {803-810},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On topological and algebraic structure of extremally disconnected semitopological groups},
url = {http://eudml.org/doc/248631},
volume = {41},
year = {2000},
}

TY - JOUR
AU - Arhangel'skii, Aleksander V.
TI - On topological and algebraic structure of extremally disconnected semitopological groups
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2000
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 41
IS - 4
SP - 803
EP - 810
AB - Starting with a very simple proof of Frol’ık’s theorem on homeomorphisms of extremally disconnected spaces, we show how this theorem implies a well known result of Malychin: that every extremally disconnected topological group contains an open and closed subgroup, consisting of elements of order $2$. We also apply Frol’ık’s theorem to obtain some further theorems on the structure of extremally disconnected topological groups and of semitopological groups with continuous inverse. In particular, every Lindelöf extremally disconnected semitopological group with continuous inverse and with square roots is countable, and every extremally disconnected topological field is discrete.
LA - eng
KW - extremally disconnected; semitopological group; order 2; Souslin number; Lindelöf space; extremally disconnected; semitopological group; order 2; Souslin number; Lindelöf space
UR - http://eudml.org/doc/248631
ER -

References

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  1. Arhangel'skii A.V., Groupes topologiques extremalement discontinus, C.R. Acad. Sci. Paris 265 (1967), 822-825. (1967) MR0222207
  2. Arhangel'skii A.V., Ponomarev V.I., Fundamentals of General Topology: Problems and Exercises, Reidel, 1984. MR0785749
  3. Efimov B.A., Absolutes of homogeneous spaces, Dokl. Akad. Nauk SSSR 179:2 (1968), 271-274. (1968) Zbl0179.27802MR0227937
  4. Frolík Z., Fixed points of maps of extremally disconnected spaces and complete Boolean Algebras, Bull. Acad. Polon. Sci., Ser. Math., Astronom., Phys. 16 (1968), 269-275. (1968) MR0233343
  5. Frolík Z., Fixed points of maps of β N , Bull. Amer. Math. Soc. 74 (1968), 187-191. (1968) MR0222847
  6. Frolík Z., Maps of extremally disconnected spaces, theory of types, and applications, General Topology and its Relations to Modern Analysis and Algebra, 3. (Proc. Conf., Kanpur, 1968), pp.131-142; Academia, Prague, 1971. MR0295305
  7. Katětov M., A theorem on mappings, Comment. Math. Univ. Carolinae 8:3 (1967), 431-433. (1967) MR0229228
  8. Malychin V.I., Extremally disconnected and close to them groups, Dokl. Akad. Nauk SSSR 220:1 (1975), 27-30. (1975) MR0382536
  9. Raimi R., Homeomorphisms and invariant measures for β N ∖ N , Duke Math. J. 33 (1966), 1-12. (1966) MR0198450
  10. Sirota S., Products of topological groups and extremal disconnectedness, Matem. Sb. 79:2 (1969), 179-192. (1969) MR0242988

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