The first formal definition of covering dimension was given by Eduard Čech, based on an earlier result of Henri Lebesgue. 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 numbern such that each point of the space belongs to, at most, n sets in the cover. A refinement of a coverC 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 has an open refinement with ply n + 1 or below. 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-dimensionalwith 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 oneopen set of this refinement.
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 has covering dimension n. A non-technical illustration of these examples is given below.
Properties
Homeomorphic spaces have the same covering dimension. That is, the covering dimension is a topological invariant.
The 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 to. Here, is the n dimensional sphere.
A normal space satisfies the inequality if and only if for every locally finite open cover of the space there exists an open cover of the space that can be represented as the union of families, where, such that each contains disjoint sets and for each and.
Karl Menger, General Spaces and Cartesian Spaces, Communications to the Amsterdam Academy of Sciences. English translation reprinted in Classics on Fractals, Gerald A.Edgar, editor, Addison-Wesley
Karl Menger, Dimensionstheorie, B.G Teubner Publishers, Leipzig.
V. V. Fedorchuk, The Fundamentals of Dimension Theory, appearing in Encyclopaedia of Mathematical Sciences, Volume 17, General Topology I, A. V. Arkhangel'skii and L. S. Pontryagin, Springer-Verlag, Berlin.
