Editing Lebesgue covering dimension (section)

==Examples==
The empty set has covering dimension −1: for any open cover of the empty set, each point of the empty set is not contained in any element of the cover, so the order of any open cover is 0.

Any given open cover of the [[unit circle]] will have a refinement consisting of a collection of [[open (topology)|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 (mathematics)|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]] <math>\mathbb{E}^n</math> has covering dimension ''n''.