On subgroups of saturated or totally bounded paratopological groups - Banakh, T.; Ravsky, S. (Algebra and Discrete Mathematics, 2003)
A paratopological group G is saturated if the inverse U
⁻¹ of each non-empty set U ⊂ G has non-empty interior. It
is shown that a [first-countable] paratopological group H is a closed
subgroup of a saturated (totally bounded) [abelian] paratopological
group if and only if H admits a continuous bijective homomorphism
onto a (totally bounded) [abelian] topological group G [such that
for each neighborhood U ⊂ H of the unit e there is a closed subset
F ⊂ G with e ∈ h
⁻¹
(F) ⊂ U]. As an application we construct a
paratopological group whose character exceeds its π-weight as well
as the character of its group reflexion. Also we present several examples of (para)topological groups which are subgroups of totally
bounded paratopological groups but fail to be subgroups of regular
totally bounded paratopological groups.