N-topological space
# N-topological space

In mathematics, an N-topological space is a set equipped with N arbitrary topologies. If $τ_1,$ $τ_2,$ ..., $τ_N$ are N topologies defined on a nonempty set X, then the N-topological space is denoted by $(X,τ_1,τ_2,...,τ_N).$ For N = 1, the structure is simply a topological space. For N = 2, the structure becomes a bitopological space introduced by J. C. Kelly.

## Example

Let X = ${x_1,$ $x_2,$ ...., $x_n}$ be any finite set. Suppose $A_r$ = ${x_1,$ $x_2,$ ..., $x_r}.$ Then the collection $τ_1$ = {φ, $A_1,$ $A_2,$ ..., $A_n$ = X} will be a topology on X. If $τ_1,$ $τ_2,$ ..., $τ_m$ be m such topologies (chain topologies) defined on X, then the structure (X, $τ_1,$ $τ_2,$ ..., $τ_m)$ is an m-topological space.

