Recall that a topological space is a set equipped with a topological structure. Well, a bitopological space is simply a set equipped with two topological structures. Unlike with bialgebras, no compatibility condition is required between these structures.

A bicontinous map is a function between bitopological spaces that is continuous with respect to each topological structure.

Bitopological spaces and bicontinuous maps form a category$BiTop$.

Remarks

It is interesting and perhaps surprising that many advanced topological notions can be described using bitopological spaces, even when you would not naïvely think that there are two topologies around. (At least, that's my vague memory of what they were good for. I think that this was in some article by Isbell.)