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 $$k_iA\subseteq k_jA$$ 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 $$X.$$