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In Bitopological spaces, Proc. London Math. Soc. (3) 13 (1963) 71–89 MR0143169, J.C. Kelly introduced the idea of bitopological spaces. Is there any paper concerning the generalization of this concept, i.e. a space with any number of topologies?

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FWIW: Among the $\sim\!50$ papers that cite this paper on MathSciNet, there seems to be none that mentions $n$-topological spaces. –  t.b. Jul 25 '11 at 3:28
    
This paper talks about n-topological spaces a little bit. –  JSchlather Jul 25 '11 at 4:10

2 Answers 2

For $n=3$ Google turns up mention of AL-Fatlawee J.K. On paracompactness in bitopological spaces and tritopological spaces, MSc. Thesis, University of Babylon (2006). Asmahan Flieh Hassan at the University of Kufa, also in Iraq, also seems to be interested in tritopological spaces and has worked with a Luay Al-Sweedy at the Univ. of Babylon. This paper by Philip Kremer makes use of tritopological spaces in a study of bimodal logics, as does this paper by J. van Benthem et al., which Kremer cites. In my admittedly limited experience with the area these are very unusual, in that they make use of a tritopological structure to study something else; virtually every other paper that I’ve seen on bi- or tritopological spaces has studied them for their own sake, usually in an attempt to extend topological notions in some reasonably nice way.

I’ve seen nothing more general than this.

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I admit that I never heard of bitopological spaces before, and I hope I don't offend anybody by asking bluntly: What would be the showcase application of these ideas? –  t.b. Jul 25 '11 at 5:01
    
I’d seen them before, but I’d never seen anything resembling an application until I ran into the logic papers that I mentioned. To be brutally honest, most of what I’d seen looked like make-work papers -- excuses to attend conferences, résumé padding, etc. -- though some was a bit more interesting, if a bit of a dead end. (Dead ends per se don’t bother me, by the way; I’m perfectly happy to investigate an idea for its own sake if it interests me.) –  Brian M. Scott Jul 25 '11 at 6:44
    
Thank you for confirming my impression. I have nothing against dead ends either, but I was a bit overwhelmed by the sheer number of papers on that topic, so I was hoping for at least some mildly interesting applications "outside the field". –  t.b. Jul 25 '11 at 7:13
    
@Theo: AFAIK one of the motivation to study bitopological spaces are asymmetric metric spaces or quasi-metric spaces, e.g. jstor.org/stable/2371174 I think they were also defined in Kelly's paper. See also math.stackexchange.com/questions/23390/… Quasi-metric generates two topologies on the given space in very natural way. –  Martin Sleziak Jul 25 '11 at 7:35
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@Theo: I am not sure, if there's anything interesting in there, but this book has a chapter named "Applications of Bitopologies." books.google.com/… –  Martin Sleziak Jul 25 '11 at 7:49

I close the question with the following answer-

On the possibility of N-topological spaces, International Journal of Mathematical Archive-3(7), 2012, 2520-2523 (http://www.ijma.info/index.php/ijma/article/view/1442)

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