In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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. A graphical illustration of a nildimensional space is a
point.
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
Refinement may refer to: Mathematics
* Equilibrium refinement, the identification of actualized equilibria in game theory
* Refinement of an equivalence relation, in mathematics
** Refinement (topology), the refinement of an open cover in mathem ...
which 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
In the mathematical field of topology, the inductive dimension of a topological space ''X'' is either of two values, the small inductive dimension ind(''X'') or the large inductive dimension Ind(''X''). These are based on the observation that, in ...
if it has a
base consisting of
clopen sets.
The three notions above agree for
separable,
metrisable spaces.
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 In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...
Hausdorff space is zero-dimensional if and only if it is totally disconnected. (See for the non-trivial direction.)
* Zero-dimensional
Polish spaces are a particularly convenient setting for
descriptive set theory
In mathematical logic, descriptive set theory (DST) is the study of certain classes of "well-behaved" subsets of the real line and other Polish spaces. As well as being one of the primary areas of research in set theory, it has applications to ot ...
. Examples of such spaces include the
Cantor space and
Baire space.
* Hausdorff zero-dimensional spaces are precisely the
subspaces of topological
powers
Powers may refer to:
Arts and media
* ''Powers'' (comics), a comic book series by Brian Michael Bendis and Michael Avon Oeming
** ''Powers'' (American TV series), a 2015–2016 series based on the comics
* ''Powers'' (British TV series), a 200 ...
where
is given the
discrete topology. Such a space is sometimes called a
Cantor cube. If is
countably infinite
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...
,
is the Cantor space.
Hypersphere
The zero-dimensional
hypersphere is a pair of points. The zero-dimensional
ball
A ball is a round object (usually spherical, but can sometimes be ovoid) with several uses. It is used in ball games, where the play of the game follows the state of the ball as it is hit, kicked or thrown by players. Balls can also be used f ...
is a point.
Notes
*
*
*
References
{{Dimension topics
Dimension
0
Descriptive set theory
Properties of topological spaces
Space, topological