In mathematics, the Lebesgue covering dimension or topological dimension of a topological space X is defined to be the minimum value of n, such that every open cover of X has an open refinement in which no point is included in more than n+1 elements. If no such minimal n exists, the space is said to be of infinite covering dimension.
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Examples
For example, consider some arbitrary open cover of the unit circle. This open cover will have a refinement consisting of a collection of open arcs. The circle has dimension 1, 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 2 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, 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 open sets, while two are in general not sufficient. The covering dimension of the disk is thus 2.
Some properties
The Lebesgue covering dimension coincides with the affine dimension 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.
Suppose that the covering dimension of a space X is less than or equal to n and A is a closed subset of X. If 
is continuous, then there is an extension of f to 
. Here, Sn is the n dimensional sphere.
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, Pavel Urysohn and Henri Lebesgue.
See also
References
Historical references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, Leipzig.
- A. R. Pears, Dimension Theory of General Spaces, (1975) Cambridge University Press. ISBN 0-521-20515-8
Modern references
- 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.
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