In general topology, a polytopological space consists of a set X {\displaystyle X} together with a family { τ i } i ∈ I {\displaystyle \{\tau _{i}\}_{i\in I}} of topologies on X {\displaystyle X} that is linearly ordered by the inclusion relation where I {\displaystyle I} 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 { k i } i ∈ I {\displaystyle \{k_{i}\}_{i\in I}} to be in non-decreasing order where k i ≤ k j {\displaystyle k_{i}\leq k_{j}} if and only if k i A ⊆ k j A {\displaystyle k_{i}A\subseteq k_{j}A} for all A ⊆ X {\displaystyle A\subseteq X} . This requires non-increasing topologies.
Formal definitions An L {\displaystyle L} -topological space ( X , τ ) {\displaystyle (X,\tau )}
is a set X {\displaystyle X} together with a monotone map τ : L → {\displaystyle \tau :L\to } Top ( X ) {\displaystyle (X)} where ( L , ≤ ) {\displaystyle (L,\leq )} is a partially ordered set and Top ( X ) {\displaystyle (X)} is the set of all possible topologies on X , {\displaystyle X,} ordered by inclusion. When the partial order ≤ {\displaystyle \leq } is a linear order then ( X , τ ) {\displaystyle (X,\tau )} is called a polytopological space. Taking L {\displaystyle L} to be the ordinal number n = { 0 , 1 , … , n − 1 } , {\displaystyle n=\{0,1,\dots ,n-1\},} an n {\displaystyle n} -topological space ( X , τ 0 , … , τ n − 1 ) {\displaystyle (X,\tau _{0},\dots ,\tau _{n-1})} can be thought of as a set X {\displaystyle X} with topologies τ 0 ⊆ ⋯ ⊆ τ n − 1 {\displaystyle \tau _{0}\subseteq \dots \subseteq \tau _{n-1}} on it. More generally a multitopological space ( X , τ ) {\displaystyle (X,\tau )} is a set X {\displaystyle X} together with an arbitrary family τ {\displaystyle \tau } 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 n {\displaystyle n} closure operators and complement operator on an arbitrary n {\displaystyle n} -topological space can together generate at most 2 ⋅ K ( n ) {\displaystyle 2\cdot K(n)} distinct operators where K ( n ) = ∑ i , j = 0 n ( i + j i ) ⋅ ( i + j j ) . {\displaystyle K(n)=\sum _{i,j=0}^{n}{\tbinom {i+j}{i}}\cdot {\tbinom {i+j}{j}}.} In 1965 the Finnish logician Jaakko Hintikka found this bound for the case n = 2 {\displaystyle n=2} and claimed it "does not appear to obey any very simple law as a function of n {\displaystyle n} ".
See also Bitopological space
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