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 ($I$ is an arbitrary index set). It is usually assumed that the topologies are in non-decreasing order, but some authors prefer to put the associated closure operators $\{k_{i}\}_{i\in I}$ in non-decreasing order (operators $k_{i}$ and $k_{j}$ satisfy $k_{i}\leq k_{j}$ if and only if $k_{i}A\subseteq k_{j}A$ for all $A\subseteq X$), 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). They subsequently became an object of study in their own right, specifically in connection with Kuratowski's closure-complement problem.

Definition

An $L$‑topological space $(X,\tau )$ is a set $X$ together with a monotone map $\tau :L\to$ Top$(X)$ where $(L,\leq )$ is a partially ordered set and Top$(X)$ is the set of all possible topologies on $X,$ ordered by inclusion. When the partial order $\leq$ 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 $X$ together with $n$ topologies $\tau _{0}\subseteq \dots \subseteq \tau _{n-1}$ on it (or $\tau _{0}\supseteq \dots \supseteq \tau _{n-1},$ depending on preference). More generally, a multitopological space $(X,\tau )$ is a set $X$ together with an arbitrary family $\tau$ of topologies on $X.$

See also

• Bitopological space