Subgroups of paratopological groups and feebly compact groups

Manuel Fernández, Mikhail G. Tkachenko
2014 Applied General Topology  
It is shown that if all countable subgroups of a semitopological group G are precompact, then G is also precompact and that the closure of an arbitrary subgroup of G is again a subgroup. We present a general method of refining the topology of a given commutative paratopological group G such that the group G with the finer topology, say, σ is again a paratopological group containing a subgroup whose closure in (G, σ) is not a subgroup. It is also proved that a feebly compact paratopological
more » ... aratopological group H is perfectly κ-normal and that every G δ -dense subspace of H is feebly compact. 2010 MSC: 22A30; 54H11 (primary); 54B05 (secondary). Proposition 5.6. Let G be a feebly compact paratopological group and D a G δ -dense subset of G. Then D is feebly compact. Proof. We know that D is feebly compact if and only if rD is feebly compact. By Lemma 5.1, rD is homeomorphic to D considered as a subspace of rG. Then, by Corollary 5.5, D is feebly compact if π(D) is a feebly compact subspace of T 0 (rG). By Lemma 2.3, T 0 (rG) is a pseudocompact topological c AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 2
doi:10.4995/agt.2014.3157 fatcat:dn4vgssku5g5hbf33ffyyaubvq