A coarse convergence group need not be precompact
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Petr Simon, Fabio Zanolin (1987)
Czechoslovak Mathematical Journal
Fric, R., Kent, Darrell C. (1981)
International Journal of Mathematics and Mathematical Sciences
Mihail G. Tkachenko (2023)
Commentationes Mathematicae Universitatis Carolinae
We construct a Hausdorff topological group such that is a precalibre of (hence, has countable cellularity), all countable subsets of are closed and -embedded in , but is not -factorizable. This solves Problem 8.6.3 from the book “Topological Groups and Related Structures" (2008) in the negative.
Peter Loth (2002)
Rendiconti del Seminario Matematico della Università di Padova
Klaas Pieter Hart, Heikki J. K. Junnila, Jan van Mill (1985)
Commentationes Mathematicae Universitatis Carolinae
L. Drewnowski, I. Labuda (1981)
Colloquium Mathematicae
A. Schinzel (1986)
Colloquium Mathematicae
Hisao Kato (1992)
Czechoslovak Mathematical Journal
Raushan Z. Buzyakova, Vladimir Vladimirovich Tkachuk, Richard Gordon Wilson (2007)
Commentationes Mathematicae Universitatis Carolinae
Given a topological property (or a class) , the class dual to (with respect to neighbourhood assignments) consists of spaces such that for any neighbourhood assignment there is with and . The spaces from are called dually . We continue the study of this duality which constitutes a development of an idea of E. van Douwen used to define -spaces. We prove a number of results on duals of some general classes of spaces establishing, in particular, that any generalized ordered space...
Vladimir Pestov, Dmitri Shakhmatov (1998)
Colloquium Mathematicae
Answering a 1982 question of Sidney A. Morris, we construct a topological group G and a subspace X such that (i) G is algebraically free over X, (ii) G is relatively free over X, that is, every continuous mapping from X to G extends to a unique continuous endomorphism of G, and (iii) G is not a varietal free topological group on X in any variety of topological groups.
J. Nienhuys (1971)
Fundamenta Mathematicae
Artur Hideyuki Tomita (2005)
Fundamenta Mathematicae
In 1990, Comfort asked Question 477 in the survey book “Open Problems in Topology”: Is there, for every (not necessarily infinite) cardinal number , a topological group G such that is countably compact for all cardinals γ < α, but is not countably compact? Hart and van Mill showed in 1991 that α = 2 answers this question affirmatively under . Recently, Tomita showed that every finite cardinal answers Comfort’s question in the affirmative, also from . However, the question has remained...
R. Dacić (1968)
Publications de l'Institut Mathématique
Otera, Daniele Ettore, Russo, Francesco G. (2010)
International Journal of Mathematics and Mathematical Sciences
Elisabetta Strickland (1987)
Mathematische Annalen
Maciej Malicki (2016)
Fundamenta Mathematicae
We study a class of abelian groups that can be defined as Polish pro-countable groups, as non-archimedean groups with a compatible two-sided invariant metric or as quasi-countable groups, i.e., closed subdirect products of countable discrete groups, endowed with the product topology. We show that for every non-locally compact, abelian quasi-countable group G there exists a closed L ≤ G and a closed, non-locally compact K ≤ G/L which is a direct product of discrete countable groups....
T.H.McH. Hanson (1971)
Semigroup forum
Clark, Bradd, Cates, Sharon (1998)
International Journal of Mathematics and Mathematical Sciences
O. Alas, I. Protasov, M. Tkačenko, V. Tkachuk, R. Wilson, I. Yaschenko (1998)
Fundamenta Mathematicae
We prove that any topological group of a non-measurable cardinality is hereditarily paracompact and strongly σ-discrete as soon as it is submaximal. Consequently, such a group is zero-dimensional. Examples of uncountable maximal separable spaces are constructed in ZFC.
Ulrich Schwanengel (1979)
Manuscripta mathematica
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