2.3 Cover Dimension

A curve in the plain is covered with three different arrangemant of disks (fig. 1 center). In the right part there are only pairs of disks with non-empty intersections, while in the center part there are triplets and in the left part even quadruplets. Thus one can arrange coverings of the curve by only one intersection of each disk with another and the cover dimension of a line is defined as .

Figure 1: The cover dimension

A set of points (fig. 1 top) can be covered with disks of a sufficient small radius so that there is no intersection between them. Their covering dimension is .

A surface (fig. 1 bottom) has covering dimension , because one needs at least two overlapping of spheres to cover the surface.

The same ideas generalize to higher dimensions so that the covering dimension say for a cube is .

A detailed definition of the covering dimension is given as follows [8].

When a set is covered with small disks, the maximal number of diks which have non-empty intersection is called the order of the cover. At the left end of the line in fig. 1 the order is , in the center it is and at the right end it is . Open covers of a set are collections of finitely open disks , such that their union covers . An open cover is called a refinement of provided for each there is such that . The order of a cover is the maximal integer , such that there are disjoint indices with . A set has covering dimension provided any open cover of admits an open refinement of order but not of order .