Coset space and orbit space of group action

Question

Assume $G $ is a topological group and $H $ is its subgroup. Is it true that the right coset space, $G/H $, and the orbit space of the action of $H $ on $G $ are homeomorphic? (I consider the quotient topologies on both sets)

Answer 1

I assume we are talking about the action given by $(h,g)\mapsto hg$. Let $orb(G)$ denote the orbit space of $G$ under the action of $H$. So what is the orbit of $g\in G$? It is simply $Hg$. Therefore literally $orb(G)=G/H$ where $G/H$ denotes the set of all right cosets. Since they are generated by the same partitioning of $G$ the quotient topologies have to agree as well.