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Categories of topological spaces with sufficiently many sequentially closed spaces

Dikran Dikranjan; Jan Pelant — 1997

Cahiers de Topologie et Géométrie Différentielle Catégoriques

Metrization criteria for compact groups in terms of their dense subgroups

Dikran Dikranjan; Dmitri Shakhmatov — 2013

Fundamenta Mathematicae

According to Comfort, Raczkowski and Trigos-Arrieta, a dense subgroup D of a compact abelian group G determines G if the restriction homomorphism Ĝ → D̂ of the dual groups is a topological isomorphism. We introduce four conditions on D that are necessary for it to determine G and we resolve the following question: If one of these conditions holds for every dense (or $G_{δ}$ -dense) subgroup D of G, must G be metrizable? In particular, we prove (in ZFC) that a compact abelian group determined by all its...

Uniform entropy vs topological entropy

Dikran Dikranjan; Hans-Peter A. Kunzi — 2015

Topological Algebra and its Applications

We discuss the connection between the topological entropy and the uniform entropy and answer several open questions from [10, 15]. We also correct several erroneous statements given in [10, 18] without proof.

Productivity of the Zariski topology on groups

Dikran N. Dikranjan; D. Toller — 2013

Commentationes Mathematicae Universitatis Carolinae

This paper investigates the productivity of the Zariski topology $ℨ_{G}$ of a group $G$ . If $𝒢 = {G_{i} ∣ i \in I}$ is a family of groups, and $G = \prod_{i \in I} G_{i}$ is their direct product, we prove that $ℨ_{G} \subseteq \prod_{i \in I} ℨ_{G_{i}}$ . This inclusion can be proper in general, and we describe the doubletons $𝒢 = {G_{1}, G_{2}}$ of abelian groups, for which the converse inclusion holds as well, i.e., $ℨ_{G} = ℨ_{G_{1}} \times ℨ_{G_{2}}$ . If $e_{2} \in G_{2}$ is the identity element of a group $G_{2}$ , we also describe the class $Δ$ of groups $G_{2}$ such that $G_{1} \times {e_{2}}$ is an elementary algebraic subset of $G_{1} \times G_{2}$ for every group $G_{1}$ . We show among others, that $Δ$ is stable...

Ordinal invariants and epimorphisms in some categories of weak Hausdorff spaces

Dikran N. Dikranjan; Eraldo Giuli — 1986

Commentationes Mathematicae Universitatis Carolinae

On convergence groups with dense coarse subgroups

Dikran N. Dikranjan; Roman Frič; Fabio Zanolin — 1987

Czechoslovak Mathematical Journal

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