# A solution to Comfort's question on the countable compactness of powers of a topological group

Fundamenta Mathematicae (2005)

- Volume: 186, Issue: 1, page 1-24
- ISSN: 0016-2736

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topArtur Hideyuki Tomita. "A solution to Comfort's question on the countable compactness of powers of a topological group." Fundamenta Mathematicae 186.1 (2005): 1-24. <http://eudml.org/doc/282747>.

@article{ArturHideyukiTomita2005,

abstract = {In 1990, Comfort asked Question 477 in the survey book “Open Problems in Topology”: Is there, for every (not necessarily infinite) cardinal number $α ≤ 2^\{\}$, a topological group G such that $G^γ$ is countably compact for all cardinals γ < α, but $G^α$ is not countably compact?
Hart and van Mill showed in 1991 that α = 2 answers this question affirmatively under $MA_\{countable\}$. Recently, Tomita showed that every finite cardinal answers Comfort’s question in the affirmative, also from $MA_\{countable\}$. However, the question has remained open for infinite cardinals.
We show that the existence of $2^\{\}$ selective ultrafilters + $2^\{\} = 2^\{<2^\{\}\}$ implies a positive answer to Comfort’s question for every cardinal $κ ≤ 2^\{\}$. Thus, it is consistent that κ can be a singular cardinal of countable cofinality. In addition, the groups obtained have no non-trivial convergent sequences.},

author = {Artur Hideyuki Tomita},

journal = {Fundamenta Mathematicae},

keywords = {countably compact topological group; topological product; selective ultrafilter; p-limit; no nontrivial convergent sequences; Martin's Axiom},

language = {eng},

number = {1},

pages = {1-24},

title = {A solution to Comfort's question on the countable compactness of powers of a topological group},

url = {http://eudml.org/doc/282747},

volume = {186},

year = {2005},

}

TY - JOUR

AU - Artur Hideyuki Tomita

TI - A solution to Comfort's question on the countable compactness of powers of a topological group

JO - Fundamenta Mathematicae

PY - 2005

VL - 186

IS - 1

SP - 1

EP - 24

AB - In 1990, Comfort asked Question 477 in the survey book “Open Problems in Topology”: Is there, for every (not necessarily infinite) cardinal number $α ≤ 2^{}$, a topological group G such that $G^γ$ is countably compact for all cardinals γ < α, but $G^α$ is not countably compact?
Hart and van Mill showed in 1991 that α = 2 answers this question affirmatively under $MA_{countable}$. Recently, Tomita showed that every finite cardinal answers Comfort’s question in the affirmative, also from $MA_{countable}$. However, the question has remained open for infinite cardinals.
We show that the existence of $2^{}$ selective ultrafilters + $2^{} = 2^{<2^{}}$ implies a positive answer to Comfort’s question for every cardinal $κ ≤ 2^{}$. Thus, it is consistent that κ can be a singular cardinal of countable cofinality. In addition, the groups obtained have no non-trivial convergent sequences.

LA - eng

KW - countably compact topological group; topological product; selective ultrafilter; p-limit; no nontrivial convergent sequences; Martin's Axiom

UR - http://eudml.org/doc/282747

ER -

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