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
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.