# gr.group theory – Left-side cosets of an open subgroup

Let $$G$$ be a topological group and $$H$$ its closed subgroup. $$K$$ and $$L$$ are open subgroups of $$G$$ and $$H$$ respectively. Let $$g_{1}, g_{2}in G$$. We assume $$Lcap g_{1}Kneq phi$$ and $$Lcap g_{2}Kneq phi$$. Then Is there an element $$xin H$$ (also $$xin L$$) such that $$Lcap g_{1}K=x(Lcap g_{2}K)$$ ?