Lebesgue covering dimension
Encyclopedia
Lebesgue covering dimension or topological dimension is one of several inequivalent notions of assigning a topological invariant dimension
to a given topological space
.
of X admits a finite open cover 
of X which refines 
in which no point is included in more than n+1 elements.
If no such minimal n exists, the space is said to be of infinite covering dimension.
has covering dimension n.
A topological space is zero-dimensional
with respect to the covering dimension if every open cover of the space has a refinement consisting of disjoint open sets so that any point in the space is contained in exactly one open set of this refinement.
Any given open cover of the unit circle
will have a refinement consisting of a collection of open arcs. The circle has dimension one, by this definition, because any such cover can be further refined to the stage where a given point x of the circle is contained in at most two open arcs. That is, whatever collection of arcs we begin with, some can be discarded or shrunk, such that the remainder still covers the circle, but with simple overlaps.
Similarly, any open cover of the unit disk in the two-dimensional plane
can be refined so that any point of the disk is contained in no more than three open sets, while two are in general not sufficient. The covering dimension of the disk is thus two.
A non-technical illustration of these examples below.
,
it was based on earlier result of Henri Lebesgue
.
Dimension
In physics and mathematics, the dimension of a 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 because only one coordinate is needed to specify a point on it...
to a given topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
.
Definition
The covering dimension of a topological space X is defined to be the minimum value of n, such that every finite open cover of X admits a finite open cover of X which refines in which no point is included in more than n+1 elements.If no such minimal n exists, the space is said to be of infinite covering dimension.
Examples
The n-dimensional Euclidean spaceEuclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
has covering dimension n.A topological space is zero-dimensional
Zero-dimensional space
In mathematics, a zero-dimensional topological 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...
with respect to the covering dimension if every open cover of the space has a refinement consisting of disjoint open sets so that any point in the space is contained in exactly one open set of this refinement.
Any given open cover of the unit circle
Unit circle
In mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, "the" unit circle is the circle of radius one centered at the origin in the Cartesian coordinate system in the Euclidean plane...
will have a refinement consisting of a collection of open arcs. The circle has dimension one, by this definition, because any such cover can be further refined to the stage where a given point x of the circle is contained in at most two open arcs. That is, whatever collection of arcs we begin with, some can be discarded or shrunk, such that the remainder still covers the circle, but with simple overlaps.
Similarly, any open cover of the unit disk in the two-dimensional plane
Plane (mathematics)
In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...
can be refined so that any point of the disk is contained in no more than three open sets, while two are in general not sufficient. The covering dimension of the disk is thus two.
A non-technical illustration of these examples below.
Properties
- Homeomorphic spaces have the same covering dimension.
- The Lebesgue covering dimension coincides with the affine dimension of a finite simplicial complexSimplicial complexIn mathematics, a simplicial complex is a topological space of a certain kind, constructed by "gluing together" points, line segments, triangles, and their n-dimensional counterparts...
; this is the Lebesgue covering theorem.
- The covering dimension of a normal spaceNormal spaceIn topology and related branches of mathematics, a normal space is a topological space X that satisfies Axiom T4: every two disjoint closed sets of X have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space...
is less than or equal to the large inductive dimensionInductive dimensionIn the mathematical field of topology, the inductive dimension of a topological space X is either of two values, the small inductive dimension ind or the large inductive dimension Ind...
.
- Covering dimension of a normal space X is
if and only if for any closed subset A of X, if 
is continuous, then there is an extension of 
to 
. Here, 
is the n dimensional sphereSphereA sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
.
- (Ostrand's theorem on colored dimension.) A normal spaceNormal spaceIn topology and related branches of mathematics, a normal space is a topological space X that satisfies Axiom T4: every two disjoint closed sets of X have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space...
satisfies the inequality 
if and only if for every locally finite open cover 
of the space 
there exists an open cover 
of the space 
which can be represented as the union of 
families 
, where 
, such that sets in each 
do not interect and 
for each 
and 
.
History
The first formal definition of covering dimension was given by Eduard ČechEduard Cech
Eduard Čech was a Czech mathematician born in Stračov, Bohemia . His research interests included projective differential geometry and topology. In 1921–1922 he collaborated with Guido Fubini in Turin...
,
it was based on earlier result of Henri Lebesgue
Henri Lebesgue
Henri Léon Lebesgue was a French mathematician most famous for his theory of integration, which was a generalization of the seventeenth century concept of integration—summing the area between an axis and the curve of a function defined for that axis...
.
Historical references
, General Spaces and Cartesian Spaces, (1926) Communications to the Amsterdam Academy of Sciences. English translation reprinted in Classics on Fractals, Gerald A.Edgar, editor, Addison-Wesley (1993) ISBN 0-201-58701-7, Dimensionstheorie, (1928) B.G Teubner Publishers, Leipzig.- A. R. Pears, Dimension Theory of General Spaces, (1975) Cambridge University Press. ISBN 0-521-20515-8
Modern references
- V.V. Fedorchuk, The Fundamentals of Dimension Theory, appearing in Encyclopaedia of Mathematical Sciences, Volume 17, General Topology I, (1993) A. V. Arkhangel'skii and L. S. Pontryagin (Eds.), Springer-Verlag, Berlin ISBN 3-540-18178-4.