arXiv subjects
Publications and source records attributed to Mikhail G. Tkachenko.
It is proved that the free Boolean topological group $B(X)$ on a Tychonoff space $X$ is Weil complete if and only if the space $X$ is Dieudonné complete. This result provides a positive answer to a question posed by the first listed author in 2015.
The famous Banach-Mazur problem, which asks if every infinite-dimensional Banach space has an infinite-dimensional separable quotient Banach space, has remained unsolved for 85 years, though it has been answered in the affirmative for reflexive Banach spaces and even Banach spaces which are duals. The analogous problem for locally convex spaces has been answered in the negative, but has been shown to be true for large classes of locally convex spaces including all non-normable Fréchet spaces. In this paper the analogous problem for topological groups is investigated. Indeed there are four natural analogues: Does every non-totally disconnected topological group have a separable quotient group which is (i) non-trivial; (ii) infinite; (iii) metrizable; (iv) infinite metrizable. All four questions are answered here in the negative. However, positive answers are proved for important classes of topological groups including (a) all compact groups; (b) all locally compact abelian groups; (c) all $σ$-compact locally compact groups; (d) all abelian pro-Lie groups; (e) all $σ$-compact pro-Lie groups; (f) all pseudocompact groups. Negative answers are proved for precompact groups.
We prove that if $H$ is a topological group such that all closed subgroups of $H$ are separable, then the product $G\times H$ has the same property for every separable compact group $G$. Let $c$ be the cardinality of the continuum. Assuming $2^{ω_1} = c$, we show that there exist: (1) pseudocompact topological abelian groups $G$ and $H$ such that all closed subgroups of $G$ and $H$ are separable, but the product $G\times H$ contains a closed non-separable $σ$-compact subgroup; (2) pseudocomplete locally convex vector spaces $K$ and $L$ such that all closed vector subspaces of $K$ and $L$ are separable, but the product $K\times L$ contains a closed non-separable $σ$-compact vector subspace.
Early this century K. H. Hofmann and S. A. Morris introduced the class of pro-Lie groups which consists of projective limits of finite-dimensional Lie groups and proved that it contains all compact groups, all locally compact abelian groups, and all connected locally compact groups and is closed under the formation of products and closed subgroups. They defined a topological group $G$ to be almost connected if the quotient group of $G$ by the connected component of its identity is compact. We show here that all almost connected pro-Lie groups as well as their continuous homomorphic images are $R$-factorizable and \textit{$ω$-cellular}, i.e.~every family of $G_δ$-sets contains a countable subfamily whose union is dense in the union of the whole family. We also prove a general result which implies as a special case that if a topological group $G$ contains a compact invariant subgroup $K$ such that the quotient group $G/K$ is an almost connected pro-Lie group, then $G$ is $R$-factorizable and $ω$-cellular. Applying the aforementioned result we show that the sequential closure and the closure of an arbitrary $G_{δ,Σ}$-set in an almost connected pro-Lie group $H$ coincide.
We prove that, under CH, any space with a regular $G_δ$-diagonal and caliber $ω_1$ is separable; a corollary of this result answers, under CH, a question of Buzyakova. For any Urysohn space $X$, we establish the inequality $|X|\le wL(X)^{sΔ_2(X)\cdot{dot(X)}}$ which represents a generalization of a theorem of Basile, Bella, and Ridderbos. We also show that if $X$ is a Hausdorff space, then $|X|\le(πχ(X)\cdot d(X))^{ot(X)\cdotψ_c(X)}$; this result implies Šapirovski{\uı}'s inequality $|X|\leπχ(X)^{c(X)\cdotψ(X)}$ which only holds for regular spaces. It is also proved that $|X|\le πχ(X)^{ot(X)\cdotψ_c(X)\cdot aL_c(X)}$ for any Hausdorff space $X$; this gives one more generalization of the famous Arhangel$^\prime$skii's inequality $|X|\le 2^{χ(X)\cdot L(X)}$.
We show that if $Y$ is a dense subspace of a Tychonoff space $X$, then $w(X)\leq nw(Y)^{Nag(Y)}$, where $Nag(Y)$ is the Nagami number of $Y$. In particular, if $Y$ is a Lindelöf $Σ$-space, then $w(X)\leq nw(Y)^ω\leq nw(X)^ω$. Better upper bounds for the weight of topological groups are given. For example, if a topological group $H$ contains a dense subgroup $G$ such that $G$ is a Lindelöf $Σ$-space, then $w(H)=w(G)\leq ψ(G)^ω$. Further, if a Lindelöf $Σ$-space $X$ generates a dense subgroup of a topological group $H$, then $w(H)\leq 2^{ψ(X)}$. Several facts about subspaces of Hausdorff separable spaces are established. It is well known that the weight of a separable Hausdorff space $X$ can be as big as $2^{2^{\mathfrak c}}$, where ${\mathfrak c}=2^ω$. We prove on the one hand that if a regular Lindelöf $Σ$-space $Y$ is a subspace of a separable Hausdorff space, then $w(Y)\leq \mathfrak c$, and the same conclusion holds for a Lindelöf $P$-space $Y$. On the other hand, we present an example of a countably compact topological group $G$ which is homeomorphic to a subspace of a separable Hausdorff space and satisfies $w(G)=2^{2^{\mathfrak c}}$, i.e. has the maximal possible weight.
A subspace Y of a separable metrizable space X is separable, but without X metrizable this is not true even If Y is a closed linear subspace of a topological vector space X. K.H. Hofmann and S.A. Morris introduced the class of pro-Lie groups which consists of projective limits of finite-dimensional Lie groups and proved that it contains all compact groups, locally compact abelian groups and connected locally compact groups and is closed under products and closed subgroups. A topological group G is almost connected if the quotient group of G by the connected component of its identity is compact. We prove that an almost connected pro-Lie group is separable iff its weight is not greater than c. It is deduced that an almost connected pro-Lie group is separable if and only if it is a subspace of a separable Hausdorff space. It is proved that a locally compact (even feathered) topological group G which is a subgroup of a separable Hausdorff topological group is separable, but the conclusion is false if it is assumed only that G is homeomorphic to a subspace of a separable Tychonoff space. Every precompact topological group of weight less than or equal to c is topologically isomorphic to a closed subgroup of a separable pseudocompact group of weight c. This implies that there is a wealth of closed nonseparable subgroups of separable pseudocompact groups. An example is presented under CH of a separable countably compact abelian group which contains a non-separable closed subgroup. It is proved that the following conditions are equivalent for an omega-narrow topological group G: (i) G is a subspace of a separable regular space; (ii) G is a subgroup of a separable topological group; (iii) G is a closed subgroup of a separable pathconnected locally pathconnected group.