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In mathematics, an N-topological space is a set equipped with N arbitrary topologies. If τ₁, τ₂, ..., τ_N are N topologies defined on a nonempty set X, then the N-topological space is denoted by (X,τ₁,τ₂,...,τ_N). 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]

Let X = {x₁, x₂, ...., x_n} be any finite set. Suppose A_r = {x₁, x₂, ..., x_r}. Then the collection τ₁ = {φ, A₁, A₂, ..., A_n = X} will be a topology on X. If τ₁, τ₂, ..., τ_m be m such topologies (chain topologies) defined on X, then the structure (X, τ₁, τ₂, ..., τ_m) is an m-topological space.

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