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A Dowker group

Klaas Pieter Hart, Heikki J. K. Junnila, Jan van Mill (1985)

Commentationes Mathematicae Universitatis Carolinae

A quest for nice kernels of neighbourhood assignments

Raushan Z. Buzyakova, Vladimir Vladimirovich Tkachuk, Richard Gordon Wilson (2007)

Commentationes Mathematicae Universitatis Carolinae

Given a topological property (or a class) 𝒫 , the class 𝒫 * dual to 𝒫 (with respect to neighbourhood assignments) consists of spaces X such that for any neighbourhood assignment { O x : x X } there is Y X with Y 𝒫 and { O x : x Y } = X . The spaces from 𝒫 * are called dually 𝒫 . We continue the study of this duality which constitutes a development of an idea of E. van Douwen used to define D -spaces. We prove a number of results on duals of some general classes of spaces establishing, in particular, that any generalized ordered space...

A relatively free topological group that is not varietal free

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.

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

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 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 M A c o u n t a b l e . Recently, Tomita showed that every finite cardinal answers Comfort’s question in the affirmative, also from M A c o u n t a b l e . However, the question has remained...

Abelian pro-countable groups and orbit equivalence relations

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....

Almost all submaximal groups are paracompact and σ-discrete

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.

Analytic Baire spaces

A. J. Ostaszewski (2012)

Fundamenta Mathematicae

We generalize to the non-separable context a theorem of Levi characterizing Baire analytic spaces. This allows us to prove a joint-continuity result for non-separable normed groups, previously known only in the separable context.

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