Polytopological space

In general topology, a polytopological space consists of a set ${\displaystyle X}$ together with a family ${\displaystyle \{\tau _{i}\}_{i\in I}}$ of topologies on ${\displaystyle X}$ that is linearly ordered by the inclusion relation where ${\displaystyle I}$ is an arbitrary index set. It is usually assumed that the topologies are in non-decreasing order.^[1][2] However some authors prefer the associated closure operators ${\displaystyle \{k_{i}\}_{i\in I}}$ to be in non-decreasing order where ${\displaystyle k_{i}\leq k_{j}}$ if and only if ${\displaystyle k_{i}A\subseteq k_{j}A}$ for all ${\displaystyle A\subseteq X}$. This requires non-increasing topologies.^[3]

Formal definitions

An ${\displaystyle L}$-topological space ${\displaystyle (X,\tau )}$ is a set ${\displaystyle X}$ together with a monotone map ${\displaystyle \tau :L\to }$ Top${\displaystyle (X)}$ where ${\displaystyle (L,\leq )}$ is a partially ordered set and Top${\displaystyle (X)}$ is the set of all possible topologies on ${\displaystyle X,}$ ordered by inclusion. When the partial order ${\displaystyle \leq }$ is a linear order then ${\displaystyle (X,\tau )}$ is called a polytopological space. Taking ${\displaystyle L}$ to be the ordinal number ${\displaystyle n=\{0,1,\dots ,n-1\},}$ an ${\displaystyle n}$-topological space ${\displaystyle (X,\tau _{0},\dots ,\tau _{n-1})}$ can be thought of as a set ${\displaystyle X}$ with topologies ${\displaystyle \tau _{0}\subseteq \dots \subseteq \tau _{n-1}}$ on it. More generally a multitopological space ${\displaystyle (X,\tau )}$ is a set ${\displaystyle X}$ together with an arbitrary family ${\displaystyle \tau }$ of topologies on it.^[2]

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).^[1] They were later used to generalize variants of Kuratowski's closure-complement problem.^[2][3] For example Taras Banakh et al. proved that under operator composition the ${\displaystyle n}$ closure operators and complement operator on an arbitrary ${\displaystyle n}$-topological space can together generate at most ${\displaystyle 2\cdot K(n)}$ distinct operators^[2] where $${\displaystyle K(n)=\sum _{i,j=0}^{n}{\tbinom {i+j}{i}}\cdot {\tbinom {i+j}{j}}.}$$In 1965 the Finnish logician Jaakko Hintikka found this bound for the case ${\displaystyle n=2}$ and claimed^[4] it "does not appear to obey any very simple law as a function of ${\displaystyle n}$".

See also

• Bitopological space