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The (dis)connectedness of products of Hausdorff spaces in the box topology

Vitalij A. Chatyrko — 2021

Commentationes Mathematicae Universitatis Carolinae

In this paper the following two propositions are proved: (a) If X α , α A , is an infinite system of connected spaces such that infinitely many of them are nondegenerated completely Hausdorff topological spaces then the box product α A X α can be decomposed into continuum many disjoint nonempty open subsets, in particular, it is disconnected. (b) If X α , α A , is an infinite system of Brown Hausdorff topological spaces then the box product α A X α is also Brown Hausdorff, and hence, it is connected. A space is Brown if...

A poset of topologies on the set of real numbers

Vitalij A. ChatyrkoYasunao Hattori — 2013

Commentationes Mathematicae Universitatis Carolinae

On the set of real numbers we consider a poset 𝒫 τ ( ) (by inclusion) of topologies τ ( A ) , where A , such that A 1 A 2 iff τ ( A 1 ) τ ( A 2 ) . The poset has the minimal element τ ( ) , the Euclidean topology, and the maximal element τ ( ) , the Sorgenfrey topology. We are interested when two topologies τ 1 and τ 2 (especially, for τ 2 = τ ( ) ) from the poset define homeomorphic spaces ( , τ 1 ) and ( , τ 2 ) . In particular, we prove that for a closed subset A of the space ( , τ ( A ) ) is homeomorphic to the Sorgenfrey line ( , τ ( ) ) iff A is countable. We study also common properties...

On countable families of sets without the Baire property

Mats AignerVitalij A. ChatyrkoVenuste Nyagahakwa — 2013

Colloquium Mathematicae

We suggest a method of constructing decompositions of a topological space X having an open subset homeomorphic to the space (ℝⁿ,τ), where n is an integer ≥ 1 and τ is any admissible extension of the Euclidean topology of ℝⁿ (in particular, X can be a finite-dimensional separable metrizable manifold), into a countable family ℱ of sets (dense in X and zero-dimensional in the case of manifolds) such that the union of each non-empty proper subfamily of ℱ does not have the Baire property in X.

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