{{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|author-link=Michiel Hazewinkel|year=1989|publisher=Kluwer Academic Publishers|page=190|isbn=9789400959941}}</ref> A graphical illustration of a zero-dimensional space is a 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 == 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 {{slink|Inductive dimension#Relationships between dimensions}}).
== 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 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 <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 countably infinite, <math>2^I</math> is the Cantor space.
== Manifolds == All points of a zero-dimensional manifold are 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}}
0 0 Category:Descriptive set theory Category:Properties of topological spaces Space, topological