Related papers: Remarks on $d$-independent topological groups

We force a classification of all the Abelian groups of cardinality at most $2^\mathfrak c$ that admit a countably compact group with a non-trivial convergent sequence. In particular, we answer (consistently) Question 24 of Dikranjan and…

Let G be an abelian group and let lambda be the smallest rank of any group whose direct sum with a free group is isomorphic to G. If lambda is uncountable, then G has lambda pairwise disjoint, non-free subgroups. There is an example where…

We show that it is consistent with ZFC that every compact group has a non-Haar-measurable subgroup. In addition, we demonstrate a natural construction, and we conjecture that this construction always produces a non-measurable subgroup of a…

We study algebraic properties on a group G such that if the discrete group G has these properties then every locally compact shift continuous topology on G with adjoined zero is either compact, or discrete. We introduce electorally flexible…

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…

We prove group existence and structure theorems in a general setting of tame topological theories. More precisely, we identify a linear/non-linear dividing line -- called topological 1-basedness -- among the class of t-minimal theories with…

Under $\mathfrak{p} = \mathfrak{c}$, we answer Question 24 of \cite{dikranjan&shakhmatov3} for cardinality ${\mathfrak c}$ , by showing that if a non-torsion Abelian group of size continuum admits a countably compact Hausdorff group…

We study topological versions of an independent set in an abelian group and a linearly independent set in a vector space, a {\em topologically independent set} in a topological group and a {\em topologically linearly independent set} in a…

We study the locally compact abelian groups in the class $\mathfrak E_{<\infty}$, that is, having only continuous endomorphisms of finite topological entropy, and in its subclass $\mathfrak E_0$, that is, having all continuous endomorphisms…

Pearl's d-separation is a foundational notion to study conditional independence between random variables. We define the topological conditional separation and we show that it is equivalent to the d-separation, extended beyond acyclic…

In this paper we investigate hereditarily normal topological groups and their subspaces. We prove that every compact subspace of a hereditarily normal topological group is metrizable. To prove this statement we first show that a…

This work is motivated by the problem of finding locally compact group topologies for piecewise full groups (a.k.a.~ topological full groups). We determine that any piecewise full group that is locally compact in the compact-open topology…

We initiate a study of asymptotic dimension for locally compact groups. This notion extends the existing invariant for discrete groups and is shown to be finite for a large class of residually compact groups. Along the way, the notion of…

We present a series of examples of nondiscrete reflexive P-groups (i.e., groups in which all $G_\delta$-sets are open) as well as noncompact reflexive $\omega$-bounded groups (in which the closure of every countable set is compact). Our…

A pointwise-elliptic subset of a topological group is one whose elements all generate relatively-compact subgroups. A connected locally compact group has a dense pointwise-elliptic subgroup if and only if it is an extension by a compact…

In this paper, we introduce Mackey functors for a t.d.l.c. group and define the cohomological dimension of this group over the Mackey category. We then compare this dimension to the rational discrete cohomological dimension defined by…

We study totally disconnected, locally compact (t.d.l.c.) groups from an algorithmic perspective. We give various approaches to defining computable presentations of t.d.l.c.\ groups, and show their equivalence. In the process, we obtain an…

A connected, locally finite graph $\Gamma$ is a Cayley--Abels graph for a totally disconnected, locally compact group $G$ if $G$ acts vertex-transitively with compact, open vertex stabilizers on $\Gamma$. Define the minimal degree of $G$ as…

The problem of the existence of non-pseudo-$\aleph_1$-compact $\mathbb R$-factorizable groups is studied. It is proved that any such group is submetrizable and has weight larger than $\omega_1$. Closely related results concerning the…

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…