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Addition theorems, D -spaces and dually discrete spaces

Ofelia Teresa AlasVladimir Vladimirovich TkachukRichard Gordon Wilson — 2009

Commentationes Mathematicae Universitatis Carolinae

A in a space X is a family 𝒪 = { O x : x X } of open subsets of X such that x O x for any x X . A set Y X is if 𝒪 ( Y ) = { O x : x Y } = X . If every neighbourhood assignment in X has a closed and discrete (respectively, discrete) kernel, then X is said to be a D -space (respectively a dually discrete space). In this paper we show among other things that every GO-space is dually discrete, every subparacompact scattered space and every continuous image of a Lindelöf P -space is a D -space and we prove an addition theorem for metalindelöf spaces which...

A quest for nice kernels of neighbourhood assignments

Raushan Z. BuzyakovaVladimir Vladimirovich TkachukRichard 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 . 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 of countable...

Connectedness and local connectedness of topological groups and extensions

Ofelia Teresa AlasMihail G. TkachenkoVladimir Vladimirovich TkachukRichard Gordon Wilson — 1999

Commentationes Mathematicae Universitatis Carolinae

It is shown that both the free topological group F ( X ) and the free Abelian topological group A ( X ) on a connected locally connected space X are locally connected. For the Graev’s modification of the groups F ( X ) and A ( X ) , the corresponding result is more symmetric: the groups F Γ ( X ) and A Γ ( X ) are connected and locally connected if X is. However, the free (Abelian) totally bounded group F T B ( X ) (resp., A T B ( X ) ) is not locally connected no matter how “good” a space X is. The above results imply that every non-trivial continuous homomorphism...

In quest of weaker connected topologies

Mihail G. TkachenkoVladimir Vladimirovich TkachukVladimir Vladimirovich UspenskijRichard Gordon Wilson — 1996

Commentationes Mathematicae Universitatis Carolinae

We study when a topological space has a weaker connected topology. Various sufficient and necessary conditions are given for a space to have a weaker Hausdorff or regular connected topology. It is proved that the property of a space of having a weaker Tychonoff topology is preserved by any of the free topological group functors. Examples are given for non-preservation of this property by “nice” continuous mappings. The requirement that a space have a weaker Tychonoff connected topology is rather...

On dense subspaces satisfying stronger separation axioms

We prove that it is independent of ZFC whether every Hausdorff countable space of weight less than c has a dense regular subspace. Examples are given of countable Hausdorff spaces of weight c which do not have dense Urysohn subspaces. We also construct an example of a countable Urysohn space, which has no dense completely Hausdorff subspace. On the other hand, we establish that every Hausdorff space of π -weight less than 𝔭 has a dense completely Hausdorff (and hence Urysohn) subspace. We show that...

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