Lebesgue covering dimension

However, not all topological spaces have this kind of "obvious" dimension, and so a precise definition is needed in such cases.

The definition proceeds by examining what happens when the space is covered by open sets.

The goal of the definition is to provide a number (an integer) that describes the space, and does not change as the space is continuously deformed; that is, a number that is invariant under homeomorphisms.

The general idea is illustrated in the diagrams below, which show a cover and refinements of a circle and a square.

The first formal definition of covering dimension was given by Eduard Čech, based on an earlier result of Henri Lebesgue.

= {Uα} is the smallest number m (if it exists) for which each point of the space belongs to at most m open sets in the cover: in other words Uα1 ∩ ⋅⋅⋅ ∩ Uαm+1 =

If no such minimal n exists, the space is said to have infinite covering dimension.

As a special case, a non-empty topological space is zero-dimensional with respect to the covering dimension if every open cover of the space has a refinement consisting of disjoint open sets, meaning any point in the space is contained in exactly one open set of this refinement.

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.