In general topology, a polytopological space consists of a set X together with a family \( {\displaystyle \{\tau _{i}\}_{i\in I}} \) of topologies on X that is linearly ordered by the inclusion relation ( I is an arbitrary index set). It is usually assumed that the topologies are in non-decreasing order,[1][2] but some authors prefer to put the associated closure operators \( {\displaystyle \{k_{i}\}_{i\in I}} \) in non-decreasing order (operators \( k_{i} \) and \( k_j \) satisfy \( {\displaystyle k_{i}\leq k_{j}} \) if and only if \( {\displaystyle k_{i}A\subseteq k_{j}A} \) for all \( A\subseteq X) \),[3] 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).[1] They subsequently became an object of study in their own right, specifically in connection with Kuratowski's closure-complement problem.[2][3]
Definition
An L L‑topological space \( (X,\tau ) \) is a set X together with a monotone map \( {\displaystyle \tau :L\to } \) Top (X) where \( (L,\leq ) \) is a partially ordered set and Top ( X ) {\displaystyle (X)} (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 \( {\displaystyle n=\{0,1,\dots ,n-1\},} \) an n‑topological space \( {\displaystyle (X,\tau _{0},\dots ,\tau _{n-1})} \) can be thought of as a set X together with n topologies \( {\displaystyle \tau _{0}\subseteq \dots \subseteq \tau _{n-1}} \) on it (or \( {\displaystyle \tau _{0}\supseteq \dots \supseteq \tau _{n-1},} \) depending on preference). More generally, a multitopological space \( (X,\tau ) \) is a set X together with an arbitrary family \( \tau \) of topologies on X.[2]
See also
Bitopological space
References
Icard, III, Thomas F. (2008). "Models of the Polymodal Provability Logic" (PDF). Master's thesis. University of Amsterdam.
Banakh, Taras; Chervak, Ostap; Martynyuk, Tetyana; Pylypovych, Maksym; Ravsky, Alex; Simkiv, Markiyan (2018). "Kuratowski Monoids of n-Topological Spaces". Topological Algebra and Its Applications. 6 (1): 1–25. doi:10.1515/taa-2018-0001.
Canilang, Sara; Cohen, Michael P.; Graese, Nicolas; Seong, Ian (2019). "The Closure-Complement-Frontier Problem in Saturated Polytopological Spaces". arXiv:1907.08203 [math.GN]: 3. arXiv:1907.08203.
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