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?
Any idea about N-topological spaces?
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3FWIW: Among the $\sim\!50$ papers that cite this paper on MathSciNet, there seems to be none that mentions $n$-topological spaces. – 2011-07-25
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0This [paper](http://zabidin.blog.umt.edu.my/files/2009/08/Mapp-and-pair-conti-on-pair-Lindelof.pdf) talks about n-topological spaces a little bit. – 2011-07-25
2 Answers
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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0I 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? – 2011-07-25
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0I’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.) – 2011-07-25
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0Thank 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". – 2011-07-25
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0@Theo: AFAIK one of the motivation to study bitopological spaces are *asymmetric metric spaces* or *quasi-metric spaces*, e.g. http://www.jstor.org/stable/2371174 I think they were also defined in Kelly's paper. See also http://math.stackexchange.com/questions/23390/examples-of-non-symmetric-distances Quasi-metric generates two topologies on the given space in very natural way. – 2011-07-25
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0@Martin: Sure, thanks. I'm aware of this example. But Kelly's paper doesn't contain anything that I'd call an application (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). – 2011-07-25
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2@Theo: I am not sure, if there's anything interesting in there, but this book has a chapter named "Applications of Bitopologies." http://books.google.com/books?id=Drp6v_Un0sMC&printsec=frontcover&dq=%22bitopological+spaces%22+applications&hl=en&ei=_R0tTsKXMsSq-Ab3ttHsDQ&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCgQ6AEwAA#v=onepage&q=applications&f=false – 2011-07-25
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0@Martin: Thanks again, I appreciate it. I'll have look if I stumble upon it next time I'm in the library. @Brian: sorry about all those pings.` – 2011-07-25
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0@Theo: No problem; I was interested to see that anything like the Dvalishvili book existed, though I doubt that I’ll try to track it down, even now that I’m retired. – 2011-07-25
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0@Brian: It seems that this discussion would better be continued here http://math.stackexchange.com/questions/53612/where-do-bitopological-spaces-naturally-occur-have-they-applications – 2011-07-25
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0@Theo Buehler, Martin, Brian: Thanks a lot for the information you have given.I was busy with the construction and now I am happy to tell you that I have finally developed some elementary theory of N-topological structures.I am little confused regarding the choice of the journal.Can you suggest me the journal appropriate for such work? – 2011-08-22
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0@Martin: Since you won't be pinged by Kamran's comment, here's a ping for you. – 2011-08-22
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0@Kamran: It is very difficult to suggest something without knowing a bit on the paper. The most obvious algorithm is to see whether one of the *recent* papers which you extend is in a journal that might accept your paper. No offense, but the short time (about a month) in which you developed this theory makes me wonder if it wouldn't be worthwhile to invest some more time and effort before trying to have it accepted. – 2011-08-22
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0@Buehler: I had some idea of this construction before I asked this question. I wanted to be sure that there were no papers introducing this idea. I followed the approach of Kelly and used the ideas of generalized metric introduced by Mustafa [Zead Mustafa and Brailey Sims, ”A New Approach to Generalized Metric Spaces”, Journal of Nonlinear and Convex Analysis, 7 (2), (2006 ). 289–297.] – 2011-08-23
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)