Zero-dimensional space: Difference between revisions
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imported>Trovatore dummy edit: my bad; the 0-D hypersphere would be embedded in R^1, not R^0. I still think this is a triviality that doesn't need to be mentioned here |
imported>D.Lazard |
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{{short description|Topological space of dimension zero}} | {{short description|Topological space of dimension zero}} | ||
{{about|zero dimension in topology|several kinds of zero space in algebra|zero object (algebra)}} | {{about|zero dimension in topology|several kinds of zero space in algebra|zero object (algebra)}}{{General geometry}} | ||
{{General geometry}} | |||
In [[mathematics]], a '''zero-dimensional topological space''' (or '''nildimensional space''') is a [[topological space]] that has dimension zero with respect to one of several inequivalent notions of assigning a [[dimension]] to a given topological space.<ref>{{cite book|url=https://books.google.com/books?id=8aHsCAAAQBAJ&q=zero-dimensional+space+math&pg=PA190|title=Encyclopaedia of Mathematics, Volume 3| first=Michiel|last=Hazewinkel|year=1989|publisher=Kluwer Academic Publishers|page=190|isbn=9789400959941}}</ref> A graphical illustration of a zero-dimensional space is a [[Point (geometry)|point]].<ref>{{cite conference|first1=Luke|last1=Wolcott|first2=Elizabeth|last2=McTernan|title=Imagining Negative-Dimensional Space|pages=637–642|book-title=Proceedings of Bridges 2012: Mathematics, Music, Art, Architecture, Culture|year=2012|editor1-first=Robert|editor1-last=Bosch|editor2-first=Douglas|editor2-last=McKenna|editor3-first=Reza|editor3-last=Sarhangi|isbn=978-1-938664-00-7|issn=1099-6702|publisher=Tessellations Publishing|location=Phoenix, Arizona, USA|url=http://bridgesmathart.org/2012/cdrom/proceedings/65/paper_65.pdf|access-date=10 July 2015}}</ref> | In [[mathematics]], a '''zero-dimensional topological space''' (or '''nildimensional space''') is a [[topological space]] that has dimension zero with respect to one of several inequivalent notions of assigning a [[dimension]] to a given topological space.<ref>{{cite book|url=https://books.google.com/books?id=8aHsCAAAQBAJ&q=zero-dimensional+space+math&pg=PA190|title=Encyclopaedia of Mathematics, Volume 3| first=Michiel|last=Hazewinkel|year=1989|publisher=Kluwer Academic Publishers|page=190|isbn=9789400959941}}</ref> A graphical illustration of a zero-dimensional space is a [[Point (geometry)|point]].<ref>{{cite conference|first1=Luke|last1=Wolcott|first2=Elizabeth|last2=McTernan|title=Imagining Negative-Dimensional Space|pages=637–642|book-title=Proceedings of Bridges 2012: Mathematics, Music, Art, Architecture, Culture|year=2012|editor1-first=Robert|editor1-last=Bosch|editor2-first=Douglas|editor2-last=McKenna|editor3-first=Reza|editor3-last=Sarhangi|isbn=978-1-938664-00-7|issn=1099-6702|publisher=Tessellations Publishing|location=Phoenix, Arizona, USA|url=http://bridgesmathart.org/2012/cdrom/proceedings/65/paper_65.pdf|access-date=10 July 2015|archive-date=26 June 2015|archive-url=https://web.archive.org/web/20150626111631/http://bridgesmathart.org/2012/cdrom/proceedings/65/paper_65.pdf|url-status=dead}}</ref> | ||
== Definition == | == Definition == | ||
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* A topological space is zero-dimensional with respect to the finite-to-finite covering dimension if every finite open cover of the space has a refinement that is a finite open cover such that any point in the space is contained in exactly one open set of this refinement. | * A topological space is zero-dimensional with respect to the finite-to-finite covering dimension if every finite open cover of the space has a refinement that is a finite open cover such that any point in the space is contained in exactly one open set of this refinement. | ||
* A topological space is zero-dimensional with respect to the [[small inductive dimension]] if it has a [[base (topology)|base]] consisting of [[clopen set]]s. | * A topological space is zero-dimensional with respect to the [[small inductive dimension]] if it has a [[base (topology)|base]] consisting of [[clopen set]]s. | ||
The three notions above agree for [[Separable space|separable]], [[metrisable space]]s | The three notions above agree for [[Separable space|separable]], [[metrisable space]]s (see {{slink|Inductive dimension#Relationships between dimensions}}). | ||
== Properties of spaces with small inductive dimension zero == | == Properties of spaces with small inductive dimension zero == | ||
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{{Dimension topics}} | {{Dimension topics}} | ||
[[Category:Dimension]] | [[Category:Dimension|0]] | ||
[[Category:Dimension theory|0]] | [[Category:Dimension theory|0]] | ||
[[Category:Descriptive set theory]] | [[Category:Descriptive set theory]] | ||
[[Category:Properties of topological spaces]] | [[Category:Properties of topological spaces]] | ||
[[Category:0 (number)|Space, topological]] | [[Category:0 (number)|Space, topological]] | ||
Latest revision as of 14:07, 19 October 2025
Template:Short description Script error: No such module "about".Template:General geometry
In mathematics, a zero-dimensional topological space (or nildimensional space) is a topological space that has dimension zero with respect to one of several inequivalent notions of assigning a dimension to a given topological space.[1] A graphical illustration of a zero-dimensional space is a point.[2]
Definition
Specifically:
- A topological space is zero-dimensional with respect to the Lebesgue covering dimension if every open cover of the space has a refinement that is a cover by disjoint open sets.
- A topological space is zero-dimensional with respect to the finite-to-finite covering dimension if every finite open cover of the space has a refinement that is a finite open cover such that any point in the space is contained in exactly one open set of this refinement.
- A topological space is zero-dimensional with respect to the small inductive dimension if it has a base consisting of clopen sets.
The three notions above agree for separable, metrisable spaces (see Template:Slink).
Properties of spaces with small inductive dimension zero
- A zero-dimensional Hausdorff space is necessarily totally disconnected, but the converse fails. However, a locally compact Hausdorff space is zero-dimensional if and only if it is totally disconnected. (See Script error: No such module "Footnotes". for the non-trivial direction.)
- Zero-dimensional Polish spaces are a particularly convenient setting for descriptive set theory. Examples of such spaces include the Cantor space and Baire space.
- Hausdorff zero-dimensional spaces are precisely the subspaces of topological powers where is given the discrete topology. Such a space is sometimes called a Cantor cube. If Template:Mvar is countably infinite, is the Cantor space.
Manifolds
All points of a zero-dimensional manifold are isolated.
Notes
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References
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