Where do bitopological spaces naturally occur? Do they have applications?
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 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.