Polytopological space

In general topology, a polytopological space consists of a set [math]\displaystyle{ X }[/math] together with a family [math]\displaystyle{ \{\tau_i\}_{i\in I} }[/math] of topologies on [math]\displaystyle{ X }[/math] that is linearly ordered by the inclusion relation ([math]\displaystyle{ I }[/math] 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 [math]\displaystyle{ \{k_i\}_{i\in I} }[/math] in non-decreasing order (operators [math]\displaystyle{ k_i }[/math] and [math]\displaystyle{ k_j }[/math] satisfy [math]\displaystyle{ k_i\leq k_j }[/math] if and only if [math]\displaystyle{ k_iA\subseteq k_jA }[/math] for all [math]\displaystyle{ A\subseteq X }[/math]), 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 [math]\displaystyle{ L }[/math]-topological space [math]\displaystyle{ (X,\tau) }[/math] is a set [math]\displaystyle{ X }[/math] together with a monotone map [math]\displaystyle{ \tau:L\to }[/math] Top[math]\displaystyle{ (X) }[/math] where [math]\displaystyle{ (L,\leq) }[/math] is a partially ordered set and Top[math]\displaystyle{ (X) }[/math] is the set of all possible topologies on [math]\displaystyle{ X, }[/math] ordered by inclusion. When the partial order [math]\displaystyle{ \leq }[/math] is a linear order, then [math]\displaystyle{ (X,\tau) }[/math] is called a polytopological space. Taking [math]\displaystyle{ L }[/math] to be the ordinal number [math]\displaystyle{ n=\{0,1,\dots,n-1\}, }[/math] an [math]\displaystyle{ n }[/math]-topological space [math]\displaystyle{ (X,\tau_0,\dots,\tau_{n-1}) }[/math] can be thought of as a set [math]\displaystyle{ X }[/math] together with [math]\displaystyle{ n }[/math] topologies [math]\displaystyle{ \tau_0\subseteq\dots\subseteq\tau_{n-1} }[/math] on it (or [math]\displaystyle{ \tau_0\supseteq\dots\supseteq\tau_{n-1}, }[/math] depending on preference). More generally, a multitopological space [math]\displaystyle{ (X,\tau) }[/math] is a set [math]\displaystyle{ X }[/math] together with an arbitrary family [math]\displaystyle{ \tau }[/math] of topologies on [math]\displaystyle{ X. }[/math]

See also

• Bitopological space