Constructing Psuedo-$\tau$-fine Precompact Groups

TL;DR

This paper explores the relationship between pseudo-$ au$-fineness and precompactness in topological groups, showing that certain combinations are possible while others are impossible, especially in groups with uncountable pseudocharacter.

Contribution

It extends the understanding of pseudo-$ au$-fine groups by analyzing their coexistence with precompactness and pseudocompactness in groups with uncountable pseudocharacter.

Findings

Precompactness can coexist with pseudo-$ au$-fineness for some bounded $ au$ in groups with uncountable pseudocharacter.

Pseudocompactness cannot coexist with pseudo-$ au$-fineness in such groups.

The coexistence depends on the boundedness of $ au$ and the pseudocharacter of the group.

Abstract

Let $\tau$ be an uncountable cardinal. The notion of a \emph{$\tau$-fine} topological group was introduced in 2021. More recently, H. Zhang et al. generalized this concept by defining pseudo-$\tau$-fine topological groups to study certain factorization properties of continuous functions on topological groups. It is known that $\tau$-fineness cannot coexist with precompactness in topological groups with uncountable character. In this paper, we investigate this problem further. We prove that, in topological groups with uncountable pseudocharacter, precompactness can coexist with pseudo-$\tau$-fineness for some bounded $\tau$ but pseudocompactness can never.