Lebesgue covering dimension

In mathematics, the Lebesgue covering dimension or topological dimension of a topological space is one of several different ways of defining the dimension of the space in a topologically invariant way.

Definition

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 contains X. The ply of a cover is the smallest number n (if it exists) such that each point of the space belongs to at most n sets in the cover. A refinement of a cover C is another cover, each of whose sets is a subset of a set in C; its ply may be smaller than, or possibly larger than, the ply of C. The covering dimension of a topological space X is defined to be the minimum value of n, such that every finite open cover C of X has a refinement with ply n + 1 or below. If no such minimal n exists, the space is said to be of infinite covering dimension.