  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.
