Lebesgue covering dimension - Definition and Overview

In mathematics, the Lebesgue covering dimension of a topological space is defined to be the minimum value of n, such that any open cover has a refinement with no point included in more than n+1 elements.

Here a refinement is a second open cover, of open sets selected from the given open cover. To illustrate the concept, consider open covers of the unit circle, by 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 arcs we begin with, enough can be discarded so that there are just simple overlaps.

The Lebesgue covering dimension gives the correct answer for the dimension of a finite simplicial complex; this is the Lebesgue covering theorem.

See also

• dimension

Example Usage of dimension