On pseudobounded and premeage paratopological groups

Abstract

Let $G$ be a paratopological group.Following F.~Lin and S.~Lin, we say that the group $G$ is pseudobounded,if for any neighborhood $U$ of the identity of $G$,there exists a natural number $n$ such that $U^n=G$.The group $G$ is $\omega$-pseudobounded,if for any neighborhood $U$ of the identity of $G$, the group $G$ is aunion of sets $U^n$, where $n$ is a natural number.The group $G$ is premeager, if $G\ne N^n$ for any nowhere dense subset $N$ of$G$ and any positive integer $n$.In this paper we investigate relations between the above classes of groups andanswer some questions posed by F. Lin, S. Lin, and S\'anchez.

On paratopological gyrogroups

“…It turns out that the Hausdorff restriction in Theorem3.5 can be removed in the latter case. To demonstrate this, we need the following theorem, which is inspired by Ravsky's result [22]. Theorem 3.12.…”

“…It turns out that in the latter situation, the T 0 constraint can be dropped. Ravsky [23] proved that a compact paratopological group is a topological group. Ellis [11], Grant [16], Brand [8], Bouziad [7], Bokalo and Guran [6], Romaguera and Sanchis [25], Kenderov et al [19], and others have all generalized the latter fact.…”

Commutative Topological Semigroups Embedded into Topological Abelian Groups

“…x }. We will prove that {gU : U ∈ B, g ∈ S } is a base for a topology making S into semitopological group, for it we will prove the conditions 1, 3, an 4 given in [15] (p. 93). It is easy to prove the conditions 1 and 4, let us see 3…”

Some three-space properties of semitopological and paratopological groups

“…The following proposition may have been proven somewhere. In [19,Theorem 3.2], the authors proved that ib(G) ≤ e(G) for each quasitopological group, and in [12] the author proved that ib(G) ≤ wl(G) for each saturated paratopological group. Therefore, we have the following question by applying Proposition 4.1.…”