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 where is an arbitrary index set. It is usually assumed that the topologies are in non-decreasing order. However some authors prefer the associated closure operators to be in non-decreasing order where if and only if for all. This requires non-increasing topologies.

Formal definitions

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 with topologies on it. More generally a multitopological space is a set together with an arbitrary family of topologies on it.

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). They were later used to generalize variants of Kuratowski's closure-complement problem. For example Taras Banakh et al. proved that under operator composition the closure operators and complement operator on an arbitrary -topological space can together generate at most distinct operators where In 1965 the Finnish logician Jaakko Hintikka found this bound for the case and claimed it "does not appear to obey any very simple law as a function of ".