Some properties involving feeble compactness, III: (Weakly) compact-bounded topological groups

We study two topological properties weaker than feeble compactness in the class of (para)topological groups, the compact-boundedness and weak compact-boundedness, both introduced by Angoa, Ortiz-Castillo and Tamariz-Mascarúa in [2]. First, given a subgroup H of a topological group G, we show how to extend these properties from the quotient space $G/H$ to G; this, in the cases when H is a compact, locally compact or (weakly) compact-bounded subgroup. Secondly, we prove the main result of this article: if a Tychonoff space X is compact-bounded and not scattered, then the free topological group $F\left(X\right)$ and the free Abelian topological group $A\left(X\right)$ admit a non-trivial metrizable quotient group; thus extending Theorem 4.7 by Leiderman and Tkachenko in [15]. Finally, we study the r-weakly compact-bounded subsets of a topological space X. We show that r-weak compact-boundedness is a productive property. Moreover, sufficient conditions are given in order for a C-compact subset of a paratopological group G to become an r-weakly compact-bounded subset. This article is part of a larger work developed in [16] and [17].