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In mathematics, an N-topological space is a set equipped with N arbitrary topologies. If τ1τ2, ..., τN are N topologies defined on a nonempty set X, then the N-topological space is denoted by (X,τ1,τ2,...,τN). For N = 1, the structure is simply a topological space. For N = 2, the structure becomes a bitopological space introduced by J. C. Kelly.[1]

Example

Let X = {x1, x2, ...., xn} be any finite set. Suppose Ar = {x1, x2, ..., xr}. Then the collection τ1 = {φA1, A2, ..., An = X} will be a topology on X. If τ1, τ2, ..., τm be m such topologies (chain topologies) defined on X, then the structure (X, τ1, τ2, ..., τm) is an m-topological space.

References

Kelly, J. C. (1963). "Bitopological spaces". Proc. London Math. Soc. 13 (3): 71–89. doi:10.1112/plms/s3-13.1.71.