# 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).

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.

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I don't know where they occur, but now they can get married here in NYC :). – gary Jul 25 '11 at 9:33
The book Bitopological Spaces: Theory, Relations with Generalized Algebraic Structures and Applications By B.P. Dvalishvili, Tbilisi, Georgia elsevier.com/wps/find/bookdescription.cws_home/704996/… books.google.com/… has a chapter called "Application of bitopologies". But it seems that I do not have sufficient background for going through them. – Martin Sleziak Jul 25 '11 at 9:48
Many hyperspace topologies (e.g. Vietoris topology) have upper and lower part, so it is another situation where the same set comes with two topologies. However, I do not know whether they were studied as bitopologies or whether there are some applications of bitopological spaces in this area. – Martin Sleziak Jul 25 '11 at 13:40
A related question was posted at MO: Bitopological spaces and algebraic topology. – Martin Sleziak Jun 20 '12 at 11:22

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.

$$\begin{gather*} d(x,y)\ge 0\\ d(x,y)=0 \Rightarrow x=y\\ d(x,z)\le d(x,y)+d(y,z) \end{gather*}$$

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.

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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.

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