Suitable sets for paratopological groups

A subset S of a paratopological group G is suitable set for G, if discrete subspace $$S\cup \{e\}$$ ∪ { e } closed, and the subgroup $$\langle S\rangle $$ ⟨ ⟩ generated by dense in G. Suitable sets topological groups were studied many authors. The aim present paper to provide start-up general investigation groups, looking what extent we can (by proving propositions) or cannot constructing examples) generalize results which hold pose few challenging questions possible future research. We shall discuss when different classes have sets. Namely, consider (in particular, countable) satisfying separation axioms, are compact-like spaces, saturated precompact) groups. Also property preserved with respect (open dense) subgroups, products extensions.