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In general topology, a polytopological space consists of a set $X$ together with a family $\{\tau _{i}\}_{i\in I}$ 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 $\{k_{i}\}_{i\in I}$ in non-decreasing order (operators $k_{i}$ and $k_{j}$ satisfy $k_{i}\leq k_{j}$ if and only if $k_{i}A\subseteq k_{j}A$ 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 $L$ -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 $n=\{0,1,\dots ,n-1\},$ an $n$ -topological space $(X,\tau _{0},\dots ,\tau _{n-1})$ can be thought of as a set $X$ together with $n$ topologies $\tau _{0}\subseteq \dots \subseteq \tau _{n-1}$ on it (or $\tau _{0}\supseteq \dots \supseteq \tau _{n-1},$ 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]

References

^ ^a ^b Icard, III, Thomas F. (2008). Models of the Polymodal Provability Logic (PDF) (Master's thesis). University of Amsterdam.
^ ^a ^b ^c 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. arXiv:1508.07703. doi:10.1515/taa-2018-0001.
^ ^a ^b Canilang, Sara; Cohen, Michael P.; Graese, Nicolas; Seong, Ian (2021). "The closure-complement-frontier problem in saturated polytopological spaces". New Zealand Journal of Mathematics. 51: 3–27. arXiv:1907.08203. MR 4374156.

[Icard2008-1] Icard, III, Thomas F. (2008). Models of the Polymodal Provability Logic (PDF) (Master's thesis). University of Amsterdam.

[Banakh2018-2] 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. arXiv:1508.07703. doi:10.1515/taa-2018-0001.

[Canilang2019-3] Canilang, Sara; Cohen, Michael P.; Graese, Nicolas; Seong, Ian (2021). "The closure-complement-frontier problem in saturated polytopological spaces". New Zealand Journal of Mathematics. 51: 3–27. arXiv:1907.08203. MR 4374156.

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 An <math>L</math>'''-topological space''' <math>(X,\tau)</math>
-is a set <math>X</math> together with a [[monotonic function|monotone map]] <math>\tau:L\to</math> '''Top'''<math>(X)</math> where <math>(L,\leq)</math> is a [[partially ordered set]] and '''Top'''<math>(X)</math> is the set of all possible topologies on <math>X,</math> ordered by inclusion. When the partial order <math>\leq</math> is a linear order, then <math>(X,\tau)</math> is called a '''polytopological space'''. Taking <math>L</math> to be the [[ordinal number#Von_Neumann_definition_of_ordinals|ordinal number]] <math>n=\{0,1,\dots,n-1\},</math> an [[N-topological space|<math>n</math>'''-topological space]]''' <math>(X,\tau_0,\dots,\tau_{n-1})</math> can be thought of as a set <math>X</math> together with <math>n</math> topologies <math>\tau_0\subseteq\dots\subseteq\tau_{n-1}</math> on it (or <math>\tau_0\supseteq\dots\supseteq\tau_{n-1},</math> depending on preference).  More generally, a '''multitopological space''' <math>(X,\tau)</math> is a set <math>X</math> together with an arbitrary family <math>\tau</math> of topologies on <math>X.</math><ref name=Banakh2018 />
+is a set <math>X</math> together with a [[monotonic function|monotone map]] <math>\tau:L\to</math> '''Top'''<math>(X)</math> where <math>(L,\leq)</math> is a [[partially ordered set]] and '''Top'''<math>(X)</math> is the set of all possible topologies on <math>X,</math> ordered by inclusion. When the partial order <math>\leq</math> is a linear order, then <math>(X,\tau)</math> is called a '''polytopological space'''. Taking <math>L</math> to be the [[ordinal number#Von_Neumann_definition_of_ordinals|ordinal number]] <math>n=\{0,1,\dots,n-1\},</math> an [[N-topological space|<math>n</math>'''-topological space''']] <math>(X,\tau_0,\dots,\tau_{n-1})</math> can be thought of as a set <math>X</math> together with <math>n</math> topologies <math>\tau_0\subseteq\dots\subseteq\tau_{n-1}</math> on it (or <math>\tau_0\supseteq\dots\supseteq\tau_{n-1},</math> depending on preference).  More generally, a '''multitopological space''' <math>(X,\tau)</math> is a set <math>X</math> together with an arbitrary family <math>\tau</math> of topologies on <math>X.</math><ref name=Banakh2018 />
 ==See also==