Polytopological space
In general topology, a polytopological space consists of a set together with a family of topologies on that is linearly ordered by the inclusion relation ( 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 in non-decreasing order (operators and satisfy if and only if for all ),[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[edit]
An -topological space is a set together with a monotone map Top where is a partially ordered set and Top is the set of all possible topologies on ordered by inclusion. When the partial order is a linear order, then is called a polytopological space. Taking to be the ordinal number an -topological space can be thought of as a set together with topologies on it (or depending on preference). More generally, a multitopological space is a set together with an arbitrary family of topologies on [2]
See also[edit]
References[edit]
- ^ 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.