polytopological space

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 where $I$ is an arbitrary index set. It is usually assumed that the topologies are in non-decreasing order.{{cite thesis

| last1 = Icard, III

| first1 = Thomas F.

| title = Models of the Polymodal Provability Logic

| date = 2008

| type = Master's thesis

| url = https://www.illc.uva.nl/Research/Publications/Reports/MoL-2008-06.text.pdf

| publisher = University of Amsterdam }}{{cite journal

| last1 = Banakh

| first1 = Taras

| last2 = Chervak

| first2 = Ostap

| last3 = Martynyuk

| first3 = Tetyana

| last4 = Pylypovych

| first4 = Maksym

| last5 = Ravsky

| first5 = Alex

| last6 = Simkiv

| first6 = Markiyan

| title = Kuratowski Monoids of $n$ -Topological Spaces

| journal = Topological Algebra and Its Applications

| date = 2018

| volume = 6

| issue = 1

| pages = 1–25

| doi = 10.1515/taa-2018-0001

| doi-access = free

| arxiv = 1508.07703 }} However some authors prefer the associated closure operators $\{k_i\}_{i\in I}$ 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.{{cite journal

| last1 = Canilang | first1 = Sara

| last2 = Cohen | first2 = Michael P.

| last3 = Graese | first3 = Nicolas

| last4 = Seong | first4 = Ian

| arxiv = 1907.08203

| journal = New Zealand Journal of Mathematics

| mr = 4374156

| pages = 3–27

| doi = 10.53733/151

| doi-access = free

| title = The closure-complement-frontier problem in saturated polytopological spaces

| volume = 51

| year = 2021}}

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 $n=\{0,1,\dots,n-1\},$ an $n$ -topological space $(X,\tau_0,\dots,\tau_{n-1})$ can be thought of as a set $X$ with topologies $\tau_0\subseteq\dots\subseteq\tau_{n-1}$ 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 $2\cdot K(n)$ distinct operators where $K(n)=\sum_{i,j=0}^n\tbinom{i+j}{i} \cdot \tbinom{i+j}{j}.$ In 1965 the Finnish logician Jaakko Hintikka found this bound for the case $n=2$ and claimed{{cite journal

| last1 = Hintikka | first1 = Jaakko

| journal = Fundamenta Mathematicae

| mr = 0195034

| pages = 97–106

| title = A closure and complement result for nested topologies

| volume = 57

| year = 1965

| doi = 10.4064/fm-57-1-97-106

| url = https://bibliotekanauki.pl/articles/1381954 }} it "does not appear to obey any very simple law as a function of $n$ ".

References

Category:Topology

polytopological space

Formal definitions

History

See also

References