I am interested where bitopological spaces occur in various parts of mathematics (i.e., what are natural examples of bitopological spaces stemming from various areas of mathematics, not from the studying bitopological spaces for their own sake.)
I would also like to know where bitopological spaces have some applications in various parts in mathematics. I quote from the discussion which motivated me to ask about them (I am quoting Theo B.): "a reasonable definition of an application: a result that doesn't mention the objects of study in the statement but uses them in the proof" (edit T.B.: This is a paraphrase of Paul Balmer's strict applications mentioned by him e.g. in his ICM talk, p.2).
The discussion in the comments to Brian M. Scott's answer here motivated me to ask about bitopological spaces.
I am not sure to which extent my background is important, but I never studied bitopological spaces, although I have read two papers on quasi-metric spaces, which are a special case.