On the quasi-component of pseudocompact abelian groups

Selective sequential pseudocompactness

We say that a topological space X is selectively sequentially pseudocompact (SSP for short) if for every sequence (U_n) of non-empty open subsets of X, one can choose a point x_n in U_n for every n in such a way that the sequence (x_n) has a convergent subsequence. We show that the class of SSP spaces is closed under taking arbitrary products and continuous images, contains the class of all dyadic spaces and forms a proper subclass of the class of strongly pseudocompact spaces introduced recently by García-Ferreira and Ortiz-Castillo. We investigate basic properties of this new class and its r…

Compact-like abelian groups without non-trivial quasi-convex null sequences

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 of groups: (a) bounded precompact abelian groups; (b) minimal abelian groups; (c) totally minimal abelian groups; (d) ω-bounded abelian groups. We also provide examples of minimal abelian groups with this property, and show that there exists a minimal pseudocompact abelian group with the same property; furthermore, under Martin's Axiom, the group may be chose…

Pontryagin duality in the class of precompact Abelian groups and the Baire property

We present a wide class of reflexive, precompact, non-compact, Abelian topological groups $G$ determined by three requirements. They must have the Baire property, satisfy the \textit{open refinement condition}, and contain no infinite compact subsets. This combination of properties guarantees that all compact subsets of the dual group $G^\wedge$ are finite. We also show that many (non-reflexive) precompact Abelian groups are quotients of reflexive precompact Abelian groups. This includes all precompact almost metrizable groups with the Baire property and their products. Finally, given a compac…

稠密伪紧与稠密连通空间的刻画

The characterizations of dense-pseudocompact and dense-connected spaces

本文定义了稠密拓扑性质，并分别给出了Tychonoff空间下稠密伪紧、稠密连通以及稠密超连通空间的等价刻画，同时推广至拓扑群等结构。

Precompact abelian groups and topological annihilators

For a compact Hausdorff abelian group K and its subgroup H, one defines the g-closure g(H) of H in K as the subgroup consisting of $χ\in K$ such that $χ(a_n)\longrightarrow 0$ in T=R/Z for every sequence {a_n} in $\hat K$ (the Pontryagin dual of K) that converges to 0 in the topology that H induces on $\hat K$. We prove that every countable subgroup of a compact Hausdorff group is g-closed, and thus give a positive answer to two problems of Dikranjan, Milan and Tonolo. We also show that every g-closed subgroup of a compact Hausdorff group is realcompact. The techniques developed in the paper a…

Long colimits of topological groups II: Free groups and vector spaces

Topological properties of the free topological group and the free abelian topological group on a space have been thoroughly studied since the 1940s. In this paper, we study the free topological $\mathbb{R}$-vector space $V(X)$ on $X$. We show that $V(X)$ is a quotient of the free abelian topological group on $[-1,1]\times X$, and use this to prove topological vector space analogues of existing results for free topological groups on pseudocompact spaces. As an application, we show that certain families of subspaces of $V(X)$ satisfy the so-called $\textit{algebraic colimit property}$ defined in…

准紧阿贝尔群的商群的密度与权

Densities and Weights of Quotients of Precompact Abelian Groups

针对Banach-Mazur问题的拓扑群版本，研究准紧阿贝尔群商群的密度特征与权的最小基数及可能权值集合，给出相关基数结论、子集存在性结果，并构造反例解决了一个公开问题。

Making group topologies with, and without, convergent sequences

(1) Every infinite, Abelian compact (Hausdorff) group K admits 2^|K|-many dense, non-Haar-measurable subgroups of cardinality |K|. When K is nonmetrizable, these may be chosen to be pseudocompact. (2) Every infinite Abelian group G admits a family A of 2^2^|G|-many pairwise nonhomeomorphic totally bounded group topologies such that no nontrivial sequence in G converges in any of the topologies T in A. (For some G one may arrange w(G,T) < 2^|G| for some T in A.) (3) Every infinite Abelian group $G$ admits a family B of 2^2^|G|-many pairwise nonhomeomorphic totally bounded group topologies, with…

Compatible group topologies on a locally quasi-convex abelian group and the Mackey group problem

For a locally quasi-convex (lqc) abelian group $G$, we give the first description of all compatible group topologies on $G$ and apply this result to the Mackey group problem for lqc groups. We characterize lqc abelian groups which are Mackey groups or admit a Mackey group topology and provide a characterization of two Mackey groups whose product is Mackey. We obtain the first characterization of locally convex spaces which are Mackey groups.

Maximally almost periodic groups and respecting properties

For a Tychonoff space $X$, denote by $\mathfrak{P}$ the family of topological properties $\mathcal{P}$ of being a convergent sequence or being a compact, sequentially compact, countably compact, pseudocompact and functionally bounded subset of $X$, respectively. A maximally almost periodic $(MAP$) group $G$ respects $\mathcal{P}$ if $\mathcal{P}(G)=\mathcal{P}(G^+)$, where $G^+$ is the group $G$ endowed with the Bohr topology. We study relations between different respecting properties from $\mathfrak{P}$ and show that the respecting convergent sequences (=the Schur property) is the weakest one…

On the Mackey problem for free abelian topological groups

Recently Außenhofer and the author independently have shown that the free abelian topological group $A(\mathbf{s})$ over a convergent sequence $\mathbf{s}$ does not admit the strongest compatible locally quasi-convex group topology that gives the first example of a locally quasi-convex abelian group without a Mackey group topology. In this note we considerably extend this example by showing that the free abelian topological group $A(X)$ over a non-discrete zero-dimensional metrizable space $X$ does not have a Mackey group topology. In particular, for every countable non-discrete metrizable spa…

Quasi-convex density and determining subgroups of compact abelian groups

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 the circle group, we show that the set of all characters which send X into U has the same size as G^*. (Here, w(G) denotes the weight of G.) A subgroup D of G determines G if the restriction homomorphism G^* --> D^* is an isomorphism between G^* and D^*. We prove that w(G) = min {|D|: D is a subgroup of G that determines G} for every infinite …

A locally quasi-convex abelian group without Mackey topology

We give the first example of a locally quasi-convex (even countable reflexive and $k_ω$) abelian group $G$ which does not admit the strongest compatible locally quasi-convex group topology. Our group $G$ is the Graev free abelian group $A_G(\mathbf{s})$ over a convergent sequence $\mathbf{s}$.

Locally compact abelian groups admitting non-trivial quasi-convex null sequences

In this paper, we show that for every locally compact abelian group G, the following statements are equivalent: (i) G contains 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; (ii) one of the subgroups {g \in G : 2g=0 and {g \in G : 3g=0} is open in G; (iii) G contains an open compact subgroup of the form Z_2^κor Z_3^κfor some cardinal κ.

局部紧阿贝尔群上相容局部拟凸拓扑的偏序集的大小

The size of the poset of compatible locally quasi-convex topologies on locally compact abelian groups

该研究确定了局部紧阿贝尔群上相容局部拟凸拓扑偏序集的大小，证明非预紧群的偏序集包含幂集序同构副本，回答了相关公开问题并给出精确基数公式。