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

Artur 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 -