Kuratowski monoids of $n$-topological spaces

Abstract

Generalizing the famous 14-set closure-complement Theorem of Kuratowski from 1922, we prove that for a set $X$ endowed with $n$ pairwise comparable topologies $τ_1\subset\dots\subsetτ_n$, by repeated application of the operations of complement and closure in the topologies $τ_1,\dots,τ_n$ to a subset $A\subset X$ we can obtains at most $2K(n)=2\sum_{i,j=0}^n\binom{i+j}{i}\binom{i+j}{j}$ distinct sets.