Spectral representations of topological groups and near-openly generated groups

TL;DR

This paper introduces near-openly generated groups, a new class within topological groups, and explores their properties, characterizations, and relationships with other well-known classes using inverse spectra and game theory.

Contribution

It defines near-openly generated groups and establishes their properties, including stability under subgroups, quotients, and completions, along with characterizations via inverse spectra and topological games.

Findings

Near-openly generated groups include almost connected pro-Lie groups.

Spaces of continuous functions $C_p(X)$ are near-openly generated.

Characterizations of near-openly generated groups are provided using inverse spectra and game theory.

Abstract

Near-openly generated groups are introduced. It is a topological and multiplicative subclass of $\mathbb R$-factorizable groups. Dense and open subgroups, quotients and Raikov completion of a near-openly generated group are near-openly generated. Almost connected pro-Lie groups, lindel\" off almost metrizable groups and the spaces $C_p(X)$ of all continuous real-valued functions on a Tychonoff space $X$ with pointwise convergence topology are near-openly generated. We provide characterizations of near-openly generated groups using methods of inverse spectra and topological game theory.