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?
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 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)