In measure theoretic language there is a notion of matched pair of locally compact (l.c.) groups due to Baaj-Skandalis-Vaes. A pair $(G\_{1}, G\_{2})$ is called a matched pair of l.c. groups if there exists a l.c. group $G$ such that, $G\_{1}$, $G\_{2}$ are closed subgroups of $G$, intersection of $G\_{1}$ and $G\_{2}$ is trivial inside $G$ and the complement of $G\_{1}G\_{2}$ is of measure zero in $G$. Does there exists any notion which replace the last condition by some topological criterion? 



Matched pair of locally compact groups
I suppose you could ask that the complement of $G\_1G\_2$ is nowhere dense, or more generally a [meagre set](https://en.wikipedia.org/wiki/Meagre_set). But whether this notion is appropriate or not really depends on what application you have in mind.


Also, unless I am missing something, isn't every pair of locally compact groups is a matched pair? Just take $G = G\_1 \times G\_2$.


