On finite powers of countably compact groups

Artur Hideyuki Tomita

Commentationes Mathematicae Universitatis Carolinae (1996)

  • Volume: 37, Issue: 3, page 617-626
  • ISSN: 0010-2628

Abstract

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We will show that under M A c o u n t a b l e for each k ∈ ℕ there exists a group whose k -th power is countably compact but whose 2 k -th power is not countably compact. In particular, for each k ∈ ℕ there exists l ∈ [ k , 2 k ) and a group whose l -th power is countably compact but the l + 1 -st power is not countably compact.

How to cite

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Tomita, Artur Hideyuki. "On finite powers of countably compact groups." Commentationes Mathematicae Universitatis Carolinae 37.3 (1996): 617-626. <http://eudml.org/doc/247889>.

@article{Tomita1996,
abstract = {We will show that under $\{M\hspace\{-1.8pt\}A\hspace\{0.2pt\}\}_\{countable\}$ for each $k \in \mathbb \{N\}$ there exists a group whose $k$-th power is countably compact but whose $2^k$-th power is not countably compact. In particular, for each $k \in \mathbb \{N\}$ there exists $l \in [k,2^k)$ and a group whose $l$-th power is countably compact but the $l+1$-st power is not countably compact.},
author = {Tomita, Artur Hideyuki},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {countable compactness; $\{M\hspace\{-1.8pt\}A\hspace\{0.2pt\}\}_\{countable\}$; topological groups; finite powers; countable compactness; MA; finite powers of topological groups},
language = {eng},
number = {3},
pages = {617-626},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On finite powers of countably compact groups},
url = {http://eudml.org/doc/247889},
volume = {37},
year = {1996},
}

TY - JOUR
AU - Tomita, Artur Hideyuki
TI - On finite powers of countably compact groups
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1996
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 37
IS - 3
SP - 617
EP - 626
AB - We will show that under ${M\hspace{-1.8pt}A\hspace{0.2pt}}_{countable}$ for each $k \in \mathbb {N}$ there exists a group whose $k$-th power is countably compact but whose $2^k$-th power is not countably compact. In particular, for each $k \in \mathbb {N}$ there exists $l \in [k,2^k)$ and a group whose $l$-th power is countably compact but the $l+1$-st power is not countably compact.
LA - eng
KW - countable compactness; ${M\hspace{-1.8pt}A\hspace{0.2pt}}_{countable}$; topological groups; finite powers; countable compactness; MA; finite powers of topological groups
UR - http://eudml.org/doc/247889
ER -

References

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  1. Comfort W.W., Topological Groups, Handbook of Set-Theoretic Topology (K. Kunen and J. Vaughan, eds.), North-Holland, 1984, pp. 1143-1263. Zbl1071.54019MR0776643
  2. Comfort W.W., Problems on topological groups and other homogeneous spaces, Open Problems in Topology (J. van Mill and G. M. Reed, eds.), North-Holland, 1990, pp. 311-347. MR1078657
  3. Comfort W.W., Hofmann K.H., Remus D., Topological groups and semigroups, Recent Progress in General Topology (M. Hušek and J. van Mill, eds.), Elsevier Science Publishers, 1992, pp. 57-144. Zbl0798.22001MR1229123
  4. Comfort W.W., Ross K.A., Pseudocompactness and uniform continuity in topological groups, Pacific J. Math. 16.3 (1966), 483-496. (1966) Zbl0214.28502MR0207886
  5. van Douwen E.K., The product of two countably compact topological groups, Trans. Amer. Math. Soc. 262 (Dec. 1980), 417-427. (Dec. 1980) Zbl0453.54006MR0586725
  6. Hajnal A., Juhász I., A separable normal topological group need not be Lindelöf, Gen. Top. and its Appl. 6 (1976), 199-205. (1976) MR0431086
  7. Hart K.P., van Mill J., A countably compact H such that H × H is not countably compact, Trans. Amer. Math. Soc. 323 (Feb. 1991), 811-821. (Feb. 1991) MR0982236
  8. Tkachenko M.G., Countably compact and pseudocompact topologies on free abelian groups, Izvestia VUZ. Matematika 34 (1990), 68-75. (1990) Zbl0714.22001MR1083312
  9. Tomita A.H., Between countable and sequential compactness in free abelian group, preprint. 
  10. Tomita A.H., A group under M A c o u n t a b l e whose square is countably compact but whose cube is not, preprint. 
  11. Tomita A.H., The Wallace Problem: a counterexample from M A c o u n t a b l e and p -compactness, to appear in Canadian Math. Bulletin. 
  12. Weiss W., Versions of Martin's Axiom, Handbook of Set-Theoretic Topology (K. Kunen and J. Vaughan, eds.), North-Holland, 1984, pp. 827-886. Zbl0571.54005MR0776638

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