Polytopological space


In general topology, a polytopological space consists of a set together with a family of topologies on that is linearly ordered by the inclusion relation. It is usually assumed that the topologies are in non-decreasing order, but some authors prefer to put the associated closure operators in non-decreasing order, 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. They subsequently became an object of study in their own right, specifically in connection with Kuratowski's closure-complement problem.

Definition

An ‑topological space
is a set together with a monotone map Top where is a partially ordered set and Top is the set of all possible topologies on ordered by inclusion. When the partial order is a linear order, then is called a polytopological space. Taking to be the ordinal number an ‑topological space can be thought of as a set together with topologies on it. More generally, a multitopological space is a set together with an arbitrary family of topologies on