Spectral representations of topological groups and near-openly generated groups

Near-openly generated groups are introduced. They form a topological and multiplicative subclass of $\mathbb R$-factorizable groups. Dense and open subgroups, quotients and the Raikov completion of a near-openly generated group are near-openly generated. Almost connected pro-Lie groups, Lindelöf almost metrizable groups and the spaces $C_p(X)$ of all continuous real-valued functions on a Tychonoff space 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.
Bibliography: 24 titles.