Countable compactness and finite powers of topological groups without convergent sequences

We show under MA countable that for every positive integer n there exists a topological group G without non-trivial convergent sequences such that G n is countably compact but G n+1 is not. This answers the finite case of Comfort's Question 477 in the Open Problems in Topology. We also show und...

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Vydané v:Topology and its applications Ročník 146; s. 527 - 538
Hlavný autor: Tomita, A.H.
Médium: Journal Article
Jazyk:English
Vydavateľské údaje: Elsevier B.V 2005
Predmet:
Countably compact
Finite power
Linear independence
Martin's Axiom
Suitable set
Topological group
Finite power
Countably compact
Topological group
Suitable set
Martin's Axiom
Linear independence
.
ISSN:0166-8641, 1879-3207
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Shrnutí:We show under MA countable that for every positive integer n there exists a topological group G without non-trivial convergent sequences such that G n is countably compact but G n+1 is not. This answers the finite case of Comfort's Question 477 in the Open Problems in Topology. We also show under MA countable +2 < c = c that there are 2 c non-homeomorphic group topologies as above if n⩾2. We apply the construction on suitable sets, answering the finite case of a question of D. Dikranjan on the productivity of suitability and in a topological game defined by Bouziad.
ISSN:0166-8641
1879-3207
DOI:10.1016/j.topol.2003.10.008