Polytopological space explained

X together with a family \{\tau _{i\}} _{i\in} of topologies on X that is linearly ordered by the inclusion relation where I 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 \{k _{i\}} _{i\in} to be in non-decreasing order where k _{i\leq} k _{j} if and only if k _{iA\subseteq} k _{jA} for all A\subseteq X . This requires non-increasing topologies.^[3]

Formal definitions

An

L -topological space (X,\tau) is a set X together with a monotone map \tau:L\to Top(X) where (L,\leq) is a partially ordered set and Top(X) is the set of all possible topologies on X, ordered by inclusion. When the partial order \leq is a linear order then (X,\tau) is called a polytopological space. Taking L to be the ordinal number n = \{ 0 , 1 , ... ,n -1 \}, an n -topological space (X,\tau _{0,...,\tau} _{n-1} ) can be thought of as a set X with topologies \tau _{0\subseteq...\subseteq\tau} _{n-1} on it. More generally a multitopological space (X,\tau) is a set X together with an arbitrary family \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 closure operators and complement operator on an arbitrary n -topological space can together generate at most 2 ⋅ K(n) distinct operators where $$K(n)=\backslash sum\_^n\backslash tbinom\; \backslash cdot\; \backslash tbinom.$$In 1965 the Finnish logician Jaakko Hintikka found this bound for the case n =2 and claimed^[4] it “does not appear to obey any very simple law as a function of n .”

See also

• Bitopological space

Notes and References

1. Icard, III . Thomas F. . Models of the Polymodal Provability Logic . 2008 . Master's thesis . University of Amsterdam .
2. Banakh . Taras . Chervak . Ostap . Martynyuk . Tetyana . Pylypovych . Maksym . Ravsky . Alex . Simkiv . Markiyan . Kuratowski Monoids of n -Topological Spaces . Topological Algebra and Its Applications . 2018 . 6 . 1 . 1–25 . 10.1515/taa-2018-0001 . free . 1508.07703 .
3. Canilang . Sara . Cohen . Michael P. . Graese . Nicolas . Seong . Ian . 1907.08203 . New Zealand Journal of Mathematics . 4374156 . 3–27 . 10.53733/151 . free . The closure-complement-frontier problem in saturated polytopological spaces . 51 . 2021.
4. Hintikka . Jaakko . Fundamenta Mathematicae . 0195034 . 97-106 . A closure and complement result for nested topologies . 57 . 1965 .