In general topology, a polytopological space consists of a set
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.
[1][2] However some authors prefer the associated closure operators
to be in non-decreasing order where
This requires non-increasing topologies.
-topological space
together with a monotone map
Top
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.
to be the ordinal number
-topological space
can be thought of as a set
More generally a multitopological space
together with an arbitrary family
[2] 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 were later used to generalize variants of Kuratowski's closure-complement problem.
[2][3] 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[2] where
In 1965 the Finnish logician Jaakko Hintikka found this bound for the case
and claimed[4] it "does not appear to obey any very simple law as a function of