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 ${\tau }_1\subset \cdots \subset {\tau }_n$, by repeated application of the operations of complement and closure in the topologies ${\tau }_1,\dots ,{\tau }_n$ to a subset $A\subset X$ we can obtains at most $2K(n)=2{\sum }_{i,j=0}^n\left(\genfrac{}{}{0ex}{}{i+j}{i}\right)\left(\genfrac{}{}{0ex}{}{i+j}{j}\right)$ distinct sets.

20 pages

References in corpus (1)

• Kuratowski's Theorem for Two Closure Operators