Lebesgue covering dimension
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In mathematics, the Lebesgue covering dimension or topological dimension of a topological space is defined to be the minimum value of n, such that any open cover has a refinement in which no point is included in more than n+1 elements. In this context, a refinement is a second open cover such every set of the second open cover is a subset of some set in the first open cover. It is named after Henri Lebesgue, although it was independently arrived at by a number of contemporaneous mathematicians.
For example, consider some arbitrary open cover of the unit circle. This open cover will consist of a collection of open arcs. The circle has dimension 1, by this definition, because any such cover can be refined to the stage where a given point x of the circle is contained in at most 2 arcs. That is, whatever collection of arcs we begin with, some can be broken up, and others discarded, such that in the remainder still covers the circle,but with simple overlaps.
Similarly, consider the unit disk in the two-dimensional plane. It is not hard to visualize that any open cover can be refined so that any point of the disk is contained in no more than three sets.
The Lebesgue covering dimension coincides with the affine dimension of a finite simplicial complex; this is the Lebesgue covering theorem.
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Some unusual topological constructions
The definition of the Lebesgue covering dimension can be used to build some unusual topological sets, such as the Sierpinski carpet. A construction can proceed as follows.
Consider, for example, a finite open covering for the two-dimensional unit disk. This covering can always be refined so that no point in the disk belongs to more than three sets. Fixing this covering, remove all of the points in the disk that belong to three sets. Depending on the refinement, this will leave possibly one or more holes in the disk. The remaining object is again two-dimensional, and again has a finite open cover. The process of selecting a cover and refining, and then punching out holes can be repeated, ad infinitum. The resulting object is homeomorphic to the Sierpinski carpet. What is curious about this construction is that the carpet has a Lebesgue covering dimension of one, and not two. Given any open covering of the carpet, one can always find a refinement such that every point belongs to at most two sets. The proof of this is essentially by contradiction: were there a covering which required membership to three sets, then the affected area would have been punched out during the construction phase. As open covers are at most countable, every such case is handled during construction. Similar constructions can be performed in higher dimensions; the three-dimensional analogue is called the Menger sponge. Curiously, the Lebesgue covering dimension of the Menger sponge is again one.
The Menger sponge has some additional curious properties. It is the universal curve. By this we mean that any possible one-dimensional curve (embedded in any number of dimensions) is homeomorphic to a subset of the Menger sponge. In a more restricted sense, any possible one-dimensional object embedded in the two-dimensional plane is homeomorphic to a subset of the Sierpinski carpet. Note that by curve we mean any object of Lebesgue dimension one; this includes trees and graphs with an arbitrary (countable) number of edges, vertices and closed loops.
History
The idea of topological dimension first became a topic of considerable interest in the early 20th century. The core ideas were independently arrived at and published by Karl Menger, L. E. J. Brouwer, Paul Urysohn and Henri Lebesgue.
See also
References
- 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, Lepzig.