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.
Bitopological spaces and bicontinuous maps form a category .
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.)
B. Dvalishvili, Bitopological Spaces: Theory, Relations with Generalized Algebraic Structures and Applications , Elsevier Amsterdam 2005.
Peter Johnstone, Collapsed Toposes as Bitopological Spaces , pp.19-35 in Categorical Topology , World Scientific Singapore 1989.
J. C. Kelly, Bitopological spaces , Proc. London Math. Soc. 13 no.3 (1963) pp.71-89.
O. K. Klinke, A. Jung, A. Moshier, A bitopological point-free approach to compactications (2011). (preprint)
R. Kopperman, Asymmetry and duality in topology , Topology Appl. 66 no.1 (1995) pp.1-39.