Polytopological space

In general topology, a polytopological space consists of a set $X$ together with a family $\{{\tau }_i{\}}_{i\in I}$ of topologies on $X$ that is linearly ordered by the inclusion relation where $I$ is an arbitrary index set. It is usually assumed that the topologies are in non-decreasing order. However some authors prefer the associated closure operators $\{k_i{\}}_{i\in I}$ to be in non-decreasing order where $k_i\le k_j$ if and only if $k_{i}A\subseteq k_{j}A$ for all $A\subseteq X$. This requires non-increasing topologies.

Formal definitions

An $L$-topological space $(X,\tau )$ is a set $X$ together with a monotone map $\tau :L\to$ Top$(X)$ where $(L,\le )$ is a partially ordered set and Top$(X)$ is the set of all possible topologies on $X,$ ordered by inclusion. When the partial order $\le$ is a linear order then $(X,\tau )$ is called a polytopological space. Taking $L$ to be the ordinal number $n=\{0,1,\dots ,n-1\},$ an $n$-topological space $(X,{\tau }_0,\dots ,{\tau }_{n-1})$ can be thought of as a set $X$ with topologies ${\tau }_0\subseteq \dots \subseteq {\tau }_{n-1}$ on it. More generally a multitopological space $(X,\tau )$ is a set $X$ together with an arbitrary family $\tau$ of topologies on it.

History

Polytopological spaces were introduced in 2008 by the philosopher Thomas Icard for the purpose of defining a topological model of Japaridze's polymodal logic (GLP). They were later used to generalize variants of Kuratowski's closure-complement problem. For example Taras Banakh et al. proved that under operator composition the $n$ closure operators and complement operator on an arbitrary $n$-topological space can together generate at most $2\cdot K(n)$ distinct operators where $$K(n)=\sum _{i,j=0}^n(\genfrac{}{}{0ex}{}{i+j}{i})\cdot (\genfrac{}{}{0ex}{}{i+j}{j}).$$In 1965 the Finnish logician Jaakko Hintikka found this bound for the case $n=2$ and claimed it "does not appear to obey any very simple law as a function of $n$".

See also

• Bitopological space