Dmitri Shakhmatov

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Let G be a topological group with the identity element e. Given a space X, we denote by Cp(X, G) the group of all continuous functions from X to G endowed with the topology of pointwise convergence, and we say that X is: (a) G-regular if, for each closed set F ⊆ X and every point x ∈ X \ F , there exist f ∈ Cp(X, G) and g ∈ G \ {e} such that f (x) = g and f(More)
If a discrete subset S of a topological group G with the identity 1 generates a dense subgroup of G and S ∪ {1} is closed in G, then S is called a suitable set for G. We apply Michael's selection theorem to offer a direct, self-contained, purely topological proof of the result of Hofmann and Morris [8] on the existence of suitable sets in locally compact(More)
For an abelian topological group G, let G denote the dual group of all continuous characters endowed with the compact open topology. Given a closed subset X of an infinite compact abelian group G such that w(X) < w(G), and an open neighbourhood U of 0 in T, we show that |{χ ∈ G : χ(X) ⊆ U }| = | G|. (Here, w(G) denotes the weight of G.) A subgroup D of G(More)
Let I be an infinite set, {Gi : i ∈ I} be a family of (topological) groups and G = i∈I Gi be its direct product. For J ⊆ I, pJ : G → j∈J Gj denotes the projection. We say that a subgroup H of G is: (i) uniformly controllable in G provided that for every finite set J ⊆ I there exists a finite set K ⊆ I such that pJ (H) = pJ (H ∩ i∈K Gi); (ii) controllable in(More)
We give a necessary and sufficient condition, in terms of a certain reflection principle, for every unconditionally closed subset of a group G to be algebraic. As a corollary, we prove that this is always the case when G is a direct product of an Abelian group with a direct product (sometimes also called a direct sum) of a family of countable groups. This(More)
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