# The Transversality on locally pseudocompact groups

## TL;DR

This paper investigates the existence of transversal topologies on various classes of locally pseudocompact, precompact, and compact groups, providing new results and characterizations that address open problems in topological group theory.

## Contribution

It proves that all locally pseudocompact, connected groups satisfy CSP and characterizes classes of locally compact groups admitting transversal topologies, solving longstanding open problems.

## Key findings

- Locally pseudocompact, connected groups satisfy CSP.
- Existence of transversal topologies on certain locally compact groups is characterized.
- Partial answers to open problems by Dikranjan, Tkachenko, and Yaschenko.

## Abstract

Two non-discrete Hausdorff group topologies $\tau, \delta$ on a group $G$ are called {\it transversal} if the least upper bound $\tau\vee \delta$ of $\tau$ and $\delta$ is the discrete topology. In this paper, we discuss the existence of transversal group topologies on locally pseudocompact, locally precompact or locally compact groups. We prove that each locally pseudocompact, connected topological group satisfies CSP, which gives an affirmative answer to a problem posed by Dikranjan, Tkachenko and Yaschenko in 2006. For a compact normal subgroup $K$ of a locally compact totally disconnected group $G$, if $G$ admits a transversal group topology then $G/K$ admits a transversal group topology, which give a partial answer again to a problem posed by Dikranjan, Tkachenko and Yaschenko in 2006. Moreover, we characterize some classes of locally compact groups that admit transversal group topologies.