mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the Lebesgue covering dimension or topological dimension of a

topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...

is one of several different ways of defining the

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 coo ...

of the space in a topologically invariant way.

Informal discussion

For ordinary

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...

s, the Lebesgue covering dimension is just the ordinary Euclidean dimension: zero for points, one for lines, two for planes, and so on. However, not all topological spaces have this kind of "obvious"

, and so a precise definition is needed in such cases. The definition proceeds by examining what happens when the space is covered by

open set In mathematics, an open set is a generalization of an Interval (mathematics)#Definitions_and_terminology, open interval in the real line. In a metric space (a Set (mathematics), set with a metric (mathematics), distance defined between every two ...

s. In general, a topological space ''X'' can be covered by open sets, in that one can find a collection of open sets such that ''X'' lies inside of their union. The covering dimension is the smallest number ''n'' such that for every cover, there is a refinement in which every point in ''X'' lies in the

intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their ...

of no more than ''n'' + 1 covering sets. This is the gist of the formal definition below. The goal of the definition is to provide a number (an

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

) that describes the space, and does not change as the space is continuously deformed; that is, a number that is invariant under

homeomorphism In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function ...

s. The general idea is illustrated in the diagrams below, which show a cover and refinements of a circle and a square.

Formal definition

The first formal definition of covering dimension was given by

Eduard Čech Eduard Čech (; 29 June 1893 – 15 March 1960) was a Czech mathematician. His research interests included projective differential geometry and topology. He is especially known for the technique known as Stone–Čech compactification (in topo ...

, based on an earlier result of

Henri Lebesgue Henri Léon Lebesgue (; ; June 28, 1875 – July 26, 1941) was a French mathematician known for his Lebesgue integration, theory of integration, which was a generalization of the 17th-century concept of integration—summing the area between an ...

. A modern definition is as follows. An

open cover In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a family 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_\alpha\su ...

of a topological space is a family of

s such that their union is the whole space,

\cup_\alpha

= . The order or ply of an open cover

\mathfrak A

= is the smallest number (if it exists) for which each point of the space belongs to at most open sets in the cover: in other words _₁ ∩ ⋅⋅⋅ ∩ _₊₁ =

\emptyset

for ₁, ..., ₊₁ distinct. A refinement of an open cover

\mathfrak A

= is another open cover

\mathfrak B

= , such that each is contained in some . The covering dimension of a topological space is defined to be the minimum value of such that every finite open cover

\mathfrak A

of ''X'' has an open refinement

\mathfrak B

with order + 1. The refinement

\mathfrak B

can always be chosen to be finite. Thus, if is finite, _₁ ∩ ⋅⋅⋅ ∩ _₊₂ =

\emptyset

for ₁, ..., ₊₂ distinct. If no such minimal exists, the space is said to have infinite covering dimension. As a special case, a non-empty 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, meaning any point in the space is contained in exactly one open set of this refinement.

Examples

The empty set has covering dimension −1: for any open cover of the empty set, each point of the empty set is not contained in any element of the cover, so the order of any open cover is 0. Any given open cover of the

unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...

will have a refinement consisting of a collection of

open Open or OPEN may refer to: Music * Open (band), Australian pop/rock band * The Open (band), English indie rock band * ''Open'' (Blues Image album), 1969 * ''Open'' (Gerd Dudek, Buschi Niebergall, and Edward Vesala album), 1979 * ''Open'' (Go ...

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 mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1: :D_1(P) = \.\, The closed unit disk around ''P'' is the set of points whose d ...

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. More generally, the ''n''-dimensional

\mathbb^n

has covering dimension ''n''.

Properties

Homeomorphic In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function betw ...

spaces have the same covering dimension. That is, the covering dimension is a

topological invariant In topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Alternatively, a topological property is a proper class of topological space ...

. *The covering dimension of a normal space ''X'' is

\le n

if and only if for any

closed subset In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a cl ...

''A'' of ''X'', if

f:A\rightarrow S^n

is continuous, then there is an extension of

f

g:X\rightarrow S^n

. Here,

S^n

is the ''n''-dimensional sphere. * Ostrand's theorem on covering dimension. If is a normal topological space and

\mathfrak A

= is a locally finite cover of of order ≤ + 1, then, for each 1 ≤ ≤ + 1, there exists a family of pairwise disjoint open sets

\mathfrak B

= shrinking

\mathfrak A

, i.e. _, ⊆ , and together covering .

Relationships to other notions of dimension

*For a paracompact space , the covering dimension can be equivalently defined as the minimum value of , such that every open cover

\mathfrak A

of (of any size) has an open refinement

\mathfrak B

with order + 1. In particular, this holds for all metric spaces. *Lebesgue covering theorem. The Lebesgue covering dimension coincides with the affine dimension of a finite

simplicial complex In mathematics, a simplicial complex is a structured Set (mathematics), set composed of Point (geometry), points, line segments, triangles, and their ''n''-dimensional counterparts, called Simplex, simplices, such that all the faces and intersec ...

. *The covering dimension of a

normal space Normal(s) or The Normal(s) may refer to: Film and television * Normal (2003 film), ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * Normal (2007 film), ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keit ...

is less than or equal to the large inductive dimension. *The covering dimension of a

paracompact In mathematics, a paracompact space is a topological space in which every open cover has an open Cover (topology)#Refinement, refinement that is locally finite collection, locally finite. These spaces were introduced by . Every compact space is par ...

Hausdorff space

X

is greater or equal to its cohomological dimension (in the sense of sheaves),Godement 1973, II.5.12, p. 236 that is, one has

H^i(X,A) = 0

for every sheaf

A

of abelian groups on

X

and every

i

larger than the covering dimension of

X

. * In a

metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...

, one can strengthen the notion of the multiplicity of a cover: a cover has ''-multiplicity'' if every -ball intersects with at most sets in the cover. This idea leads to the definitions of the

asymptotic dimension In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...

and Assouad–Nagata dimension of a space: a space with asymptotic dimension is -dimensional "at large scales", and a space with Assouad–Nagata dimension is -dimensional "at every scale".

Informal discussion

Formal definition

Examples

Properties

Relationships to other notions of dimension

See also

Notes

References

Further reading

Historical

Modern

External links