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.{{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}} A graphical illustration of a zero-dimensional space is a point.{{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}}
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.{{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 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 {{mvar|I}} is countably infinite, is the Cantor space.
Manifolds
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:Descriptive set theory