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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 ...
, the Lebesgue covering dimension or topological dimension of 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 points ...
is one of several different ways of defining the
dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
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, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
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"
dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
, 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, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suf ...
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 Union commonly refers to: * Trade union, an organization of workers * Union (set theory), in mathematics, a fundamental operation on sets Union may also refer to: Arts and entertainment Music * Union (band), an American rock group ** ''Un ...
. The covering dimension is the smallest number ''n'' such that for every cover, there is 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 ...
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 i ...
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 (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
) that describes the space, and does not change as the space is continuously deformed; that is, a number that is invariant under
homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
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 theory of integration, which was a generalization of the 17th-century concept of integration—summing the area between an axis and the curve of ...
. 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 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_\alpha\s ...
of a topological space is a family of
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suf ...
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 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 ...
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. 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 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. A graphical ...
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. 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.


Examples

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 Eucl ...
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'' (Gotthard album), 1999 * ''Open'' (Cowboy Junkies album), 2001 * ''Open'' (YF ...
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 di ...
in the two-dimensional
plane Plane(s) most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface Plane or planes may also refer to: Biology * Plane (tree) or ''Platanus'', wetland native plant * ''Planes' ...
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
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
\mathbb^n has covering dimension ''n''.


Properties

*
Homeomorphic In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
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 spaces ...
. *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 clo ...
''A'' of ''X'', if f:A\rightarrow S^n is continuous, then there is an extension of f to g:X\rightarrow S^n . Here, S^n is the ''n''-dimensional sphere. * Ostrand's theorem on colored dimension. If is a normal topological space and \mathfrak A = is a locally finite cover of of order ≤ + l, 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 Affine may describe any of various topics concerned with connections or affinities. It may refer to: * Affine, a relative by marriage in law and anthropology * Affine cipher, a special case of the more general substitution cipher * Affine comb ...
of a finite
simplicial complex In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set ...
. *The covering dimension of a
normal space In 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. Th ...
is less than or equal to the large
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 ...
. *The covering dimension of a
paracompact In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is normal, ...
Hausdorff space X is greater or equal to its
cohomological dimension In abstract algebra, cohomological dimension is an invariant of a group which measures the homological complexity of its representations. It has important applications in geometric group theory, topology, and algebraic number theory. Cohomological ...
(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 together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
, 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 metric geometry, asymptotic dimension of a metric space is a large-scale analog of Lebesgue covering dimension. The notion of asymptotic dimension was introduced by Mikhail Gromov in his 1993 monograph ''Asymptotic invariants of infinite groups ...
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".


See also

*
Carathéodory's extension theorem In measure theory, Carathéodory's extension theorem (named after the mathematician Constantin Carathéodory) states that any pre-measure defined on a given ring of subsets ''R'' of a given set ''Ω'' can be extended to a measure on the σ- ...
* Geometric set cover problem *
Dimension theory 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 coord ...
*
Metacompact space In the mathematical field of general topology, a topological space is said to be metacompact if every open cover has a point-finite open refinement. That is, given any open cover of the topological space, there is a refinement that is again an open ...
*
Point-finite collection In mathematics, a collection  \mathcal of subsets of a topological space X is said to be point-finite if every point of X lies in only finitely many members of \mathcal.. A topological space in which every open cover admits a point-finite o ...


Notes


References

* * * * * *


Further reading


Historical

*
Karl Menger Karl Menger (January 13, 1902 – October 5, 1985) was an Austrian-American mathematician, the son of the economist Carl Menger. In mathematics, Menger studied the theory of algebras and the dimension theory of low- regularity ("rough") curves a ...
, ''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) *
Karl Menger Karl Menger (January 13, 1902 – October 5, 1985) was an Austrian-American mathematician, the son of the economist Carl Menger. In mathematics, Menger studied the theory of algebras and the dimension theory of low- regularity ("rough") curves a ...
, ''Dimensionstheorie'', (1928) B.G Teubner Publishers, Leipzig.


Modern

* * 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 .


External links

* {{Dimension topics Dimension theory