Box Resolvability

TL;DR

This paper introduces the concepts of box and partial box resolvability in topological groups, proving that groups with certain properties are resolvable at various cardinalities, expanding understanding of their structure.

Contribution

It defines new notions of box and partial box resolvability in topological groups and proves their existence under specific conditions, such as containing an injective convergent sequence or being totally bounded.

Findings

Groups with an injective convergent sequence are box ω-resolvable.

Infinite totally bounded groups are partially box n-resolvable for all natural n.

Infinite totally bounded groups are box κ-resolvable for all infinite κ less than their size.

Abstract

We say that a topological group $G$ is partially box $\kappa$-resolvable if there exist a dense subset $B$ of $G$ and a subset $A $ of $G$, $|A|=\kappa$ such that the subsets $\{ aB: a\in A\}$ are pairwise disjoint. If $G=AB$ then $G$ is called box $\kappa$-resolvable. We prove two theorems. If a topological group $G$ contains an injective convergent sequence then $G$ is box $\omega$-resolvable. Every infinite totally bounded topological group $G$ is partially box $n$-resolvable for each natural number $n$, and $G$ is box $\kappa$-resolvable for each infinite cardinal $\kappa, \kappa<|G|$.