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 (${\displaystyle I}$ is an arbitrary index set). It is usually assumed that the topologies are in non-decreasing order,^[1][2] but some authors prefer to put the associated closure operators ${\displaystyle \{k_{i}\}_{i\in I}}$ in non-decreasing order (operators ${\displaystyle k_{i}}$ and ${\displaystyle k_{j}}$ satisfy ${\displaystyle k_{i}\leq k_{j}}$ if and only if ${\displaystyle k_{i}A\subseteq k_{j}A}$ for all ${\displaystyle A\subseteq X}$),^[3] in which case the topologies have to be non-increasing.

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 subsequently became an object of study in their own right, specifically in connection with Kuratowski's closure-complement problem.^[2][3]

Definition

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 ${\displaystyle X}$ together with ${\displaystyle n}$ topologies ${\displaystyle \tau _{0}\subseteq \dots \subseteq \tau _{n-1}}$ on it (or ${\displaystyle \tau _{0}\supseteq \dots \supseteq \tau _{n-1},}$ depending on preference). More generally, a multitopological space ${\displaystyle (X,\tau )}$ is a set ${\displaystyle X}$ together with an arbitrary family ${\displaystyle \tau }$ of topologies on ${\displaystyle X.}$^[2]

See also

• Bitopological space

References