Lebesgue covering dimension

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

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

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

Refinement of the cover of a circle

The left diagram shows a refinement (on the left) of a cover (on the right) of a circular line (black). Notice how in the refinement no point on the line is contained in more than two sets. Note also how the sets link to each other to form a "chain".

Refinement of the cover of a square

The bottom left is a refinement of a cover (top) of a planar shape (dark) so that all points in the shape are contained in at most three sets. The bottom right is an attempt to refine the cover so that no point would be contained in more than two sets. This fails in the intersection of set borders. Thus, a planar shape isn't "webby" or cannot be covered with "chains", but is in a sense thicker; i.e., its topological dimension must be higher than one.

Formal definition[]

The first formal definition of covering dimension was given by Eduard Čech, based on an earlier result of Henri Lebesgue.^[1]

A modern definition is as follows. An open cover of a topological space X is a family of open sets whose union includes X. The ply or order of a cover is the smallest number n (if it exists) such that each point of the space belongs to, at most, n sets in the cover. A refinement of a cover C is another cover, each of whose sets is a subset of a set in C. The covering dimension of a topological space X is defined to be the minimum value of n, such that every open cover C of X (regardless of ply) has an open refinement with ply n + 1 or less. If no such minimal n exists, the space is said to be of infinite covering dimension.

As a special case, 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.

It is often convenient to say that the covering dimension of the empty set is −1.

Examples[]

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.

More generally, the n-dimensional Euclidean space ${\displaystyle \mathbb {E} ^{n}}$ has covering dimension n.

Properties[]

• Homeomorphic spaces have the same covering dimension. That is, the covering dimension is a topological invariant.
• The Lebesgue covering dimension coincides with the of a finite simplicial complex; this is the Lebesgue covering theorem.
• The covering dimension of a normal space is less than or equal to the large inductive dimension.
• The covering dimension of a normal space X is ${\displaystyle \leq n}$ if and only if for any closed subset A of X, if ${\displaystyle f:A\rightarrow S^{n}}$ is continuous, then there is an extension of ${\displaystyle f}$ to ${\displaystyle g:X\rightarrow S^{n}}$. Here, ${\displaystyle S^{n}}$ is the n dimensional sphere.
• (Ostrand's theorem on colored dimension.) A normal space ${\displaystyle X}$ satisfies the inequality ${\displaystyle 0\leq \dim X\leq n}$ if and only if for every locally finite open cover ${\displaystyle {\mathcal {U}}=\{U_{\alpha }\}_{\alpha \in {\mathcal {A}}}}$ of the space ${\displaystyle X}$ there exists an open cover ${\displaystyle {\mathcal {V}}}$ of the space ${\displaystyle X}$ that can be represented as the union of ${\displaystyle n+1}$ families ${\displaystyle {\mathcal {V}}_{1},{\mathcal {V}}_{2},\dots ,{\mathcal {V}}_{n+1}}$, where ${\displaystyle {\mathcal {V}}_{i}=\{V_{i,\alpha }\}_{\alpha \in {\mathcal {A}}}}$, such that each ${\displaystyle {\mathcal {V}}_{i}}$ contains disjoint sets and ${\displaystyle V_{i,\alpha }\subseteq U_{\alpha }}$ for each ${\displaystyle i}$ and ${\displaystyle \alpha }$.
• The covering dimension of a paracompact Hausdorff space ${\displaystyle X}$ is greater or equal to its cohomological dimension (in the sense of sheaves),^[2] that is, one has ${\displaystyle H^{i}(X,A)=0}$ for every sheaf ${\displaystyle A}$ of abelian groups on ${\displaystyle X}$ and every ${\displaystyle i}$ larger than the covering dimension of ${\displaystyle X}$.

See also[]

• Carathéodory's extension theorem
• Geometric set cover problem
• Dimension theory
• Metacompact space
• Point-finite collection

References[]

• Godement, Roger (1973), Topologie algébrique et théorie des faisceaux, Paris: Hermann, MR 0345092
• Munkres, James R. (2000). Topology (2nd ed.). Prentice-Hall. ISBN 0-13-181629-2.

Further reading[]

Historical[]

• Karl Menger, 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
• Karl Menger, 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[]

• 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.

External links[]

• "Lebesgue dimension", Encyclopedia of Mathematics, EMS Press, 2001 [1994]