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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| Vydáno v: | Topology and its applications Ročník 146; s. 527 - 538 |
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| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Elsevier B.V
2005
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| Témata: | |
| ISSN: | 0166-8641, 1879-3207 |
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| Abstract | 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. |
|---|---|
| AbstractList | 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. |
| Author | Tomita, A.H. |
| Author_xml | – sequence: 1 givenname: A.H. surname: Tomita fullname: Tomita, A.H. email: tomita@ime.usp.br, tomita@sylow.math.sci.ehime-u.ac.jp organization: Departamento de Matemática, Instituto de Matemática e Estatı́stica, Universidade de São Paulo, Caixa Postal 66281, CEP 05315-970, São Paulo, Brazil |
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| CitedBy_id | crossref_primary_10_1016_j_topol_2015_05_076 crossref_primary_10_1016_j_topol_2011_12_025 crossref_primary_10_1016_j_topol_2024_108932 crossref_primary_10_1016_j_topol_2006_07_001 crossref_primary_10_1016_j_topol_2018_08_009 crossref_primary_10_1016_j_apal_2020_102910 crossref_primary_10_1016_j_topol_2019_02_040 crossref_primary_10_1016_j_topol_2005_01_026 |
| Cites_doi | 10.1090/S0002-9904-1967-11653-7 10.1016/0016-660X(76)90033-7 10.1016/S0166-8641(99)00047-4 10.4064/fm-66-2-185-193 10.1016/0166-8641(93)90074-N 10.4064/fm-40-1-106-112 10.1016/S0022-4049(98)00075-9 10.1016/S0166-8641(97)00206-X 10.1016/S0166-8641(98)00109-6 10.1016/S0166-8641(97)00129-6 10.2140/pjm.1975.57.403 10.1090/S0002-9947-1980-0586725-8 10.2140/pjm.1966.16.483 10.1016/0016-660X(76)90024-6 10.1016/S0166-8641(97)00190-9 10.1016/0022-4049(90)90142-5 10.1090/S0002-9947-1966-0203679-7 |
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| Keywords | Finite power Countably compact Topological group Suitable set Martin's Axiom Linear independence . |
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Algebra doi: 10.1016/0022-4049(90)90142-5 – ident: 10.1016/j.topol.2003.10.008_BIB013 – volume: 124 start-page: 131 year: 1966 ident: 10.1016/j.topol.2003.10.008_BIB024 article-title: Products of nearly compact spaces publication-title: Trans. Amer. Math. Soc doi: 10.1090/S0002-9947-1966-0203679-7 – volume: 43 start-page: 675 issue: 4 year: 2002 ident: 10.1016/j.topol.2003.10.008_BIB012 article-title: Topological games and product spaces. Topological games and product spaces publication-title: Comment. Math. Univ. Carolin – volume: 10 start-page: 221 issue: 1 year: 1985 ident: 10.1016/j.topol.2003.10.008_BIB030 article-title: On products of countably compact spaces publication-title: Topology Proc – volume: vol. 201 start-page: 327 year: 1998 ident: 10.1016/j.topol.2003.10.008_BIB022 article-title: On thin generating sets in topological groups |
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| Snippet | 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... |
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| SubjectTerms | Countably compact Finite power Linear independence Martin's Axiom Suitable set Topological group |
| Title | Countable compactness and finite powers of topological groups without convergent sequences |
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