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In general topology, a polytopological space consists of a set ${\displaystyle X}$ together with a family ${\displaystyle \{\tau _{i}\}_{i\in I}}$ of topologies on ${\displaystyle X}$ that is linearly ordered by the inclusion relation where ${\displaystyle I}$ 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 ${\displaystyle \{k_{i}\}_{i\in I}}$ to be in non-decreasing order where ${\displaystyle k_{i}\leq k_{j}}$ if and only if ${\displaystyle k_{i}A\subseteq k_{j}A}$ for all ${\displaystyle A\subseteq X}$. This requires non-increasing topologies.

Formal definitions

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

See also

• Bitopological space

References

1. ^ Icard, III, Thomas F. (2008). Models of the Polymodal Provability Logic (PDF) (Master's thesis). University of Amsterdam.
2. ^ Banakh, Taras; Chervak, Ostap; Martynyuk, Tetyana; Pylypovych, Maksym; Ravsky, Alex; Simkiv, Markiyan (2018). "Kuratowski Monoids of ${\displaystyle n}$-Topological Spaces". Topological Algebra and Its Applications. 6 (1): 1–25. arXiv:1508.07703. doi:10.1515/taa-2018-0001.
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. doi:10.53733/151. MR 4374156.
4. Hintikka, Jaakko (1965). "A closure and complement result for nested topologies". Fundamenta Mathematicae. 57: 97–106. MR 0195034.

• Topology