Related papers: Characterizing sequences for precompact group topo…

Let $G$ be a group and $\sigma, \tau$ be topological group topologies on $G$. We say that $\sigma$ is a successor of $\tau$ if $\sigma$ is strictly finer than $\tau$ and there is not a group topology properly between them. In this note, we…

We study generic properties of topological groups in the sense of Baire category. First we investigate countably infinite (discrete) groups. We extend a classical result of B. H. Neumann, H. Simmons and A. Macintyre on algebraically closed…

According to Markov, a subset of an abelian group G of the form {x in G: nx=a}, for some integer n and some element a of G, is an elementary algebraic set; finite unions of elementary algebraic sets are called algebraic sets. We prove that…

We consider precompact sequential and Fr\'echet group topologies and show that some natural constructions of such topologies always result in metrizable groups answering a question of D.~Dikranjan et al. We show that it is consistent that…

In this paper, we study some characterizations of $q$-spaces, strict $q$-spaces and strong $q$-spaces under $\omega$-balanced topological groups as follows: (1) A topological group $G$ is $\omega$-balanced and a $q$-space if and only if for…

A Hausdorff topological group $(G,\tau)$ is called locally minimal if there exists a neighborhood $U$ of 0 in $\tau$ such that $U$ fails to be a neighborhood of zero in any Hausdorff group topology on $G$ which is strictly coarser than…

In this paper, we study precompact abelian groups G that contain no sequence {x_n} such that {0} \cup {\pm x_n : n \in N} is infinite and quasi-convex in G, and x_n --> 0. We characterize groups with this property in the following classes…

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 say that a subgroup $H$ of an infinite compact Abelian group $X$ is {\it $T$-characterized} if there is a $T$-sequence $\mathbf{u} =\{u_n \}$ in the dual group of $X$ such that $H=\{x\in X: \; (u_n, x)\to 1 \}$. We show that a closed…

We show that every Abelian group satisfying a mild cardinal inequality admits a pseudocompact group topology from which all countable subgroups inherit the maximal totally bounded topology (we say that such a topology satisfies property…

For an uncountable cardinal \tau and a subset S of an abelian group G, the following conditions are equivalent: (i) |{ns:s\in S}|\ge \tau for all integers n\ge 1; (ii) there exists a group homomorphism \pi:G\to T^{2^\tau} such that \pi(S)…

A group is small if it has countably many complete $n$-types over the empty set for each natural number n. More generally, a group $G$ is weakly small if it has countably many complete 1-types over every finite subset of G. We show here…

In this paper, we pose the concepts of pre-topological groups and some generalizations of pre-topological groups. First, we systematically investigate some basic properties of pre-topological groups; in particular, we prove that each…

The Birkhoff--Kakutani Theorem asserts that a topological group is metrizable if and only if it has countable character. We develop and apply tools for the estimation of the character for a wide class of nonmetrizable topological groups. We…

The paper deals with group dualities. A group duality is simply a pair $(G, H)$ where $G$ is an abstract abelian group and $H$ a subgroup of characters defined on $G$. A group topology $\tau$ defined on $G$ is {\it compatible} with the…

\noindent The most natural group topology on $\Z$ is the discrete one. There are other well-known group topologies on $\Z$, like the $p$-adic, defined for any prime number $p$. It is also an important group topology the weak topology with…

Consider the algebraic dynamics on a torus T=G_m^n given by a matrix M in GL_n(Z). Assume that the characteristic polynomial of M is prime to all polynomials X^m-1. We show that any finite equivariant map from another algebraic dynamics…

The topological group version of the celebrated Banach-Mazur problem asks wether every infinite topological group has a non-trivial separable quotient group. It is known that compact groups have infinite separable metrizable quotient…

We prove that a large class of metrizable group topologies for subgroups of $\mathbb{R}^n$ and the completions of the subgroups are locally isometric to, respectively, metrizable group topologies for $\mathbb{Z}$ and their completions,…

Let $G$ be a finite group. Denote by $\textrm{Irr}(G)$ the set of all irreducible complex characters of $G.$ Let $\textrm{cd}(G)=\{\chi(1)\;|\;\chi\in \textrm{Irr}(G)\}$ be the set of all irreducible complex character degrees of $G$…