Topological Rough Groups

Abstract

The concept of topological group is a simple combination of the concepts of abstract group and topological space. The purpose of this paper is to combine the concepts of topological space and rough groups; called topological rough groups on an approximation space.

The topological properties of q-spaces in free topological groups

“…However, there exists a topological rough group which is not a homogeneous space, see Example 3.1. In [3], the authors proved the following proposition. Therefore, it is natural to pose the following question.…”

“…Definition 2.6. [3] A topological rough group is a rough group (G, * ) together with a topology τ on G satisfying the following two properties:…”

“…Proposition 2.7. [3] Let G be a topological rough group. Then, every rough subgroup H of G with relative topology is a topological rough subgroup.…”

“…In [3], the authors gave the following proposition. However, we find that the proof of this result has a gap when e ∈ G. Indeed, we have the following counterexample.…”

“…In [3], the authors proved that the intersection of two topological rough subgroups H 1 and H 2 of the topological rough group G is a topological rough subgroup if H 1 ∩ H 2 = H 1 ∩ H 2 . However, the condition "H 1 ∩ H 2 = H 1 ∩ H 2 " is only a sufficient condition but not a necessary, see Example 3.4.…”