Do we ever use two structures of the same type on the same underlying space? In other words, are there any areas of study that utilise spaces of the type $(X,a,b)$ where $X$ is a space and $a,b$ could be different choices of topologies, group operations, inner products, etc.?
One of the only examples I can think of is a ring which is a set equipped with two different monoid operations (with additional properties). Apart from this, which structures (e.g. topologies, metrics, norms, etc.) do we ever use more than one of for a given underlying space? Operations are one thing, but I'm particularly interested in if two topologies are ever used in tandem and what the applications would be.