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
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
The general idea is illustrated in the diagrams below, which show a cover and refinements of a circle and a square.
The first image shows a refinement (on the bottom) of a colored cover (on the top) of a black circular line. Note how in the refinement, no point on the circle is contained in more than two sets, and also how the sets link to one another to form a "chain". | |
The top half of the second image shows a cover (colored) of a planar shape (dark), where all of the shape's points are contained in anywhere from one to all four of the cover's sets. The bottom illustrates that any attempt to refine said cover such that no point would be contained in more than two sets—ultimately fails at the intersection of set borders. Thus, a planar shape is not "webby": it cannot be covered with "chains", per se. Instead, it proves to be thicker in some sense. More rigorously put, its topological dimension must be greater than 1. |
Formal definition
The first formal definition of covering dimension was given by Eduard Čech, based on an earlier result of Henri Lebesgue.[4]
A modern definition is as follows. An
As a special case, a non-empty topological space is
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
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 has covering dimension n.
Properties
- topological invariant.
- The covering dimension of a normal space X is if and only if for any closed subsetA of X, if is continuous, then there is an extension of to . Here, is the n-dimensional sphere.
- Ostrand's theorem on colored dimension. If X is a normal topological space and = {Uα} is a locally finite cover of X of order ≤ n + 1, then, for each 1 ≤ i ≤ n + 1, there exists a family of pairwise disjoint open sets i = {Vi,α} shrinking , i.e. Vi,α ⊆ Uα, and together covering X.[5]
Relationships to other notions of dimension
- For a paracompact space X, the covering dimension can be equivalently defined as the minimum value of n, such that every open cover of X (of any size) has an open refinement with order n + 1.[6] 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.
- The covering dimension of a normal space is less than or equal to the large inductive dimension.
- The covering dimension of a paracompact Hausdorff space is greater or equal to its cohomological dimension (in the sense of sheaves),[7] that is, one has for every sheaf of abelian groups on and every larger than the covering dimension of .
- In a metric space, one can strengthen the notion of the multiplicity of a cover: a cover has r-multiplicity n + 1 if every r-ball intersects with at most n + 1 sets in the cover. This idea leads to the definitions of the asymptotic dimension and Assouad–Nagata dimension of a space: a space with asymptotic dimension n is n-dimensional "at large scales", and a space with Assouad–Nagata dimension n is n-dimensional "at every scale".
See also
- Carathéodory's extension theorem
- Geometric set cover problem
- Dimension theory
- Metacompact space
- Point-finite collection
Notes
- .
- MR 0567548.
- ^ Lebesgue 1921.
- ISBN 9780821800119,
Lebesgue's discovery led later to the introduction by E. Čech of the covering dimension
. - ^ Ostrand 1971.
- MR 0482697.
- ^ Godement 1973, II.5.12, p. 236
References
- Edgar, Gerald A. (2008). "Topological Dimension". Measure, topology, and fractal geometry. Undergraduate Texts in Mathematics (Second ed.). MR 2356043.
- Engelking, Ryszard (1978). Dimension theory (PDF). North-Holland Mathematical Library. Vol. 19. Amsterdam-Oxford-New York: North-Holland. MR 0482697.
- MR 0102797.
- Hurewicz, Witold; Wallman, Henry (1941). Dimension Theory. Princeton Mathematical Series. Vol. 4. MR 0006493.
- MR 3728284.
- Ostrand, Phillip A. (1971). "Covering dimension in general spaces". General Topology and Appl. 1 (3): 209–221. MR 0288741.
Further reading
Historical
- ISBN 0-201-58701-7
- Karl Menger, Dimensionstheorie, (1928) B.G Teubner Publishers, Leipzig.
Modern
- Pears, Alan R. (1975). Dimension Theory of General Spaces. MR 0394604.
- 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 ISBN 3-540-18178-4.
External links
- "Lebesgue dimension", Encyclopedia of Mathematics, EMS Press, 2001 [1994]