Editing Zero-dimensional space

{{short description|Topological space of dimension zero}}
{{about|zero dimension in topology|several kinds of zero space in algebra|zero object (algebra)}}
{{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>

== 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 (topology)|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 (topology)|base]] consisting of [[clopen set]]s.
The three notions above agree for [[Separable space|separable]], [[metrisable space]]s.{{citation needed|reason=Please cite a proof.|date=April 2017}}{{clarify|reason=Is the agreement only for the zero dimensional-case, or for all dimensions?|date=April 2017}}

== 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 {{harv|Arhangel'skii|Tkachenko|2008|loc=Proposition 3.1.7, p.136}} for the non-trivial direction.)
* Zero-dimensional [[Polish space]]s are a particularly convenient setting for [[descriptive set theory]]. Examples of such spaces include the [[Cantor space]] and [[Baire space (set theory)|Baire space]].
* Hausdorff zero-dimensional spaces are precisely the [[Subspace topology|subspaces]] of topological [[power set|powers]] <math>2^I</math> where <math>2=\{0,1\}</math> is given the [[discrete topology]]. Such a space is sometimes called a [[Cantor cube]]. If {{mvar|I}} is [[countable set|countably infinite]], <math>2^I</math> is the Cantor space.

== Manifolds ==
All points of a zero-dimensional [[manifold]] are [[isolated point|isolated]]. 

== Notes ==
* {{cite book | last1=Arhangel'skii | first1= Alexander | author-link1 = Alexander Arhangelskii | last2 = Tkachenko | first2 = Mikhail | title=Topological Groups and Related Structures | series=Atlantis Studies in Mathematics | volume=1 | publisher=Atlantis Press | year=2008 | isbn=978-90-78677-06-2}}
* {{cite book | author=Engelking, Ryszard | title=General Topology | publisher=PWN, Warsaw | year=1977| author-link=Ryszard Engelking }}
* {{cite book | author=Willard, Stephen | title=General Topology | publisher=Dover Publications | year=2004 | isbn=0-486-43479-6}}

== References ==
{{reflist}}

{{Dimension topics}}

[[Category:Dimension]]
[[Category:Dimension theory|0]]
[[Category:Descriptive set theory]]
[[Category:Properties of topological spaces]]
[[Category:0 (number)|Space, topological]]