Completions of altered topological subgroups of \mathbb{R}^n

Weak completions of paratopological groups

Given a $T_0$ paratopological group $G$ and a class $\mathcal C$ of continuous homomorphisms of paratopological groups, we define the $\mathcal C$-$semicompletion$ $\mathcal C[G)$ and $\mathcal C$-$completion$ $\mathcal C[G]$ of the group $G$ that contain $G$ as a dense subgroup, satisfy the $T_0$-separation axiom and have certain universality properties. For special classes $\mathcal C$, we present some necessary and sufficient conditions on $G$ in order that the (semi)completions $\mathcal C[G)$ and $\mathcal C[G]$ be Hausdorff. Also, we give an example of a Hausdorff paratopological abelian…

Properness, Cauchy-indivisibility and the Weil completion of a group of isometries

In this paper we introduce a new class of metric actions on separable (not necessarily connected) metric spaces called "Cauchy-indivisible" actions. This new class coincides with that of proper actions on locally compact metric spaces and, as examples show, it may be different in general. The concept of "Cauchy-indivisibility" follows a more general research direction proposal in which we investigate the impact of basic notions in substantial results, like the impact of local compactness and connectivity in the theory of proper transformation groups. In order to provide some basic theory for t…

Simply $sm$-factorizable (para)topological groups and their completions

Let us call a (para)topological group \emph{strongly submetrizable} if it admits a coarser separable metrizable (para)topological group topology. We present a characterization of simply $sm$-factorizable (para)topo\-logical groups by means of continuous real-valued functions. We show that a (para)topo\-logical group $G$ is a simply $sm$-factorizable if and only if for each continuous function $f\colon G\to \mathbb{R}$, one can find a continuous homomorphism $φ$ of $G$ onto a strongly submetrizable (para)topological group $H$ and a continuous function $g\colon H\to \mathbb{R}$ such that $f=g\ci…

拓扑群上一致测度的一个嵌入定理

An embedding theorem for uniform measures on topological groups

本文将 Roelcke–Dierolf 嵌入定理推广到拓扑群及其中性子群商上的一致测度空间，证明相关自然映射为单射且在正锥上为拓扑嵌入。

On zero-dimensionality and the connected component of locally pseudocompact groups

A topological group is locally pseudocompact if it contains a non-empty open set with pseudocompact closure. In this note, we prove that if G is a group with the property that every closed subgroup of G is locally pseudocompact, then G_0 is dense in the component of the completion of G, and G/G_0 is zero-dimensional. We also provide examples of hereditarily disconnected pseudocompact groups with strong minimality properties of arbitrarily large dimension, and thus show that G/G_0 may fail to be zero-dimensional even for totally minimal pseudocompact groups.

序列完备的局部极小群的结构

The Structure of Sequentially Complete Locally Minimal Groups

本文研究局部极小预紧 Abel 群的精细结构，证明序列闭稠密子群连通分支的权与完备化一致，并刻画临界局部极小群及类 \CCC 中有限维紧 Abel 群。