 Lebesgue covering dimension

Lebesgue covering dimension or topological dimension is one of several inequivalent notions of assigning a topological invariant dimension to a given topological space.
Contents
Definition
The covering dimension of a topological space X is defined to be the minimum value of n, such that every finite open cover of X admits a finite open cover of X which refines 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.
Examples
The ndimensional Euclidean space has covering dimension n.
A topological space is zerodimensional 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.
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 twodimensional 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.
A nontechnical illustration of these examples below.
Properties
 Homeomorphic spaces have the same covering dimension.
 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.
 Covering dimension of a normal space X is if and only if for any closed subset A of X, if is continuous, then there is an extension of f to . Here, S^{n} is the n dimensional sphere.
 (Ostrand's theorem on colored dimension.) A normal space X satisfies the inequality if and only if for every locally finite open cover of the space X there exists an open cover of the space X which can be represented as the union of n + 1 families , where , such that sets in each do not interect and for each i and α.
History
The first formal definition of covering dimension was given by Eduard Čech, it was based on earlier result of Henri Lebesgue.
See also
 Dimension theory
 Metacompact space
 Pointfinite collection
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, AddisonWesley (1993) ISBN 0201587017
 Karl Menger, Dimensionstheorie, (1928) B.G Teubner Publishers, Leipzig.
 A. R. Pears, Dimension Theory of General Spaces, (1975) Cambridge University Press. ISBN 0521205158
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.), SpringerVerlag, Berlin ISBN 3540181784.
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