Where do bitopological spaces naturally occur? Do they have applications? 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.
 A: One of the situations where bitopological spaces occur naturally are asymmetric metric spaces or quasi-metric spaces. They are defined as metric spaces, but the symmetry in the definition of metric is omitted.
$$
\begin{gather*}
d(x,y)\ge 0\\
d(x,y)=0 \Rightarrow x=y\\
d(x,z)\le d(x,y)+d(y,z)
\end{gather*}
$$
We already had a discussion about quasi-metrics here.
Such spaces naturally bear two topologies: forward topology  $\tau_+$ generated by the sets
$$B^+(x,\varepsilon)=\{y\in X; d(x,y)<\varepsilon\}$$
backward topology $\tau_-$ generated by sets
$$B^-(x,\varepsilon)=\{y\in X; d(y,x)<\varepsilon\}$$
The papers


*

*Wilson, On quasi-metric spaces American Journal of Mathematics 53 No. 3 (1931), 675–684. 

*J.C. Kelly, Bitopological spaces, Proc. London Math. Soc. (3) 13 (1963), 71–89, MR0143169.
appear frequently as refences in works on this topic. A natural generalization to quasi-uniform spaces has been studied, too.

As far as the applications of this concept are concerned, let's have a look what some authors publishing in this area can say:
Isaac Vikram Chenchiah, Marc Oliver Rieger, Johannes Zimmer: Gradient flows in asymmetric metric spaces Nonlinear Analysis: Theory, Methods & Applications; Volume 71, Issue 11, Pages 5820–5834 
Not only applications in science and engineering suggest that
the symmetry requirement of a metric is often too restrictive; Gromov points out the limiting effects of this assumption [10, Introduction]. 
[10] M. Gromov, Metric structures for Riemannian and non-Riemannian spaces, in: Progress in Mathematics, vol. 152, Birkhäuser Boston Inc., Boston, MA, 1999, based on the 1981 French original [MR0682063 (85e:53051)], With appendices by M. Katz, P. Pansu and S. Semmes, Translated from the French by Sean Michael Bates.
J. Collins, J. Zimmer: An Asymmetric Arzela-Ascoli theorem,  Topology and its Applications Volume 154, Issue 11, 1 June 2007, Pages 2312–2322 
In the realms of applied mathematics and materials science we find many recent applications of asymmetric metric
spaces; for example, in rate-independent models for plasticity [6], shape-memory alloys [8], and models for material failure [12]. 
[6] A. Mainik, A. Mielke, Existence results for energetic models for rate-independent systems, Calc. Var. Partial Differential Equations 22 (1) (2005) 73–99. 
[8] A. Mielke, T. Roubíček, A rate-independent model for inelastic behavior of shape-memory alloys, Multiscale Model. Simul. 1 (4) (2003) 571–597 (electronic). 
[12] M.O. Rieger, J. Zimmer, Young measure flow as a model for damage, Preprint 11/05, Bath Institute for Complex Systems, Bath, UK, 2005.
A: A situation where two
topologies on the same set occur naturally are epireflections and
coreflections in the category of topological spaces. In such cases,
one of the two topologies is finer than the other one.
Perhaps it's worth mentioning that this condition also appears in some papers on
quasi-metric spaces. Namely I mean the condition called "approximate
metric axiom" (AMA) in the paper
Santanu Bhunia and Pratulananda Das: Two valued measure and
summability of double sequences in asymmetric context, Acta Mathematica Hungarica, 130 (1–2), 167–187
 and without any name in 
J. Collins, J. Zimmer: An Asymmetric Arzela-Ascoli theorem,  Topology and its Applications Volume 154, Issue 11, 1 June 2007, Pages 2312–2322 
(AMA) implies $\tau_+\prec\tau_-$. I went briefly over these two
papers and I have the feeling that some of the results would still
hold when using the condition $\tau_+\prec\tau_-$ instead of (AMA).
In my opinion, if some facts
which hold in these setting could be stated and proven in a unifying
way using bitopological spaces such that one of the topologies is
finer than the other one, I would personally prefer such
formulation (even in a paper which deals with only one of these
settings). I do not know whether bitopological spaces with this property have been studied.
