In
mathematics, a zero-dimensional topological space (or nildimensional space) is a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
that has dimension zero with respect to one of several inequivalent notions of assigning a
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
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
In mathematics, the Lebesgue covering dimension or topological dimension of a topological space is one of several different ways of defining the dimension of the space in a
topologically invariant way.
Informal discussion
For ordinary Euclidean ...
if every
open cover
In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alph ...
of the space has a
refinement 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 if it has a
base consisting of
clopen set
In topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open and closed. That this is possible may seem counter-intuitive, as the common meanings of and are antonyms, but their mathematical d ...
s.
The three notions above agree for
separable,
metrisable spaces.
Properties of spaces with small inductive dimension zero
* A zero-dimensional
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the many ...
is necessarily
totally disconnected
In topology and related branches of mathematics, a totally disconnected space is a topological space that has only singletons as connected subsets. In every topological space, the singletons (and, when it is considered connected, the empty set ...
, 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 e ...
Hausdorff space is zero-dimensional if and only if it is totally disconnected. (See for the non-trivial direction.)
* Zero-dimensional
Polish space
In the mathematical discipline of general topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named ...
s 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 oth ...
. Examples of such spaces include the
Cantor space In mathematics, a Cantor space, named for Georg Cantor, is a topological abstraction of the classical Cantor set: a topological space is a Cantor space if it is homeomorphic to the Cantor set. In set theory, the topological space 2ω is called "t ...
and
Baire space
In mathematics, a topological space X is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior.
According to the Baire category theorem, compact Hausdorff spaces and complete metric spaces are ...
.
* 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
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are ''isolated'' from each other in a certain sense. The discrete topology is the finest t ...
. Such a space is sometimes called a
Cantor cube
In mathematics, a Cantor cube is a topological group of the form ''A'' for some index set ''A''. Its algebraic and topological structures are the group direct product and product topology over the cyclic group of order 2 (which is itself given t ...
. 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
In mathematics, an -sphere or a hypersphere is a topological space that is homeomorphic to a ''standard'' -''sphere'', which is the set of points in -dimensional Euclidean space that are situated at a constant distance from a fixed point, ...
is a pair of points. The zero-dimensional
ball is a point.
Notes
*
*
*
References
{{Dimension topics
Dimension
0
Descriptive set theory
Properties of topological spaces
Space, topological