Before proceeding to discuss further extensions of theorems proved true for the straight line such as Cantor's Theorem, p. 38, or the Heine-Borel Theorem, it is desirable to explain fully what is meant by a region of the plane or higher space.

In the straight line we have only one possible linear element, the small interval or segment. If we lay such elements on one another, side by side, or overlapping, or move such an element about, it generates only one form of region, the larger interval or segment. The common part of two such segments that overlap is a segment, the parts left over in each are again segments. This makes the whole theory of regions on the straight line a comparatively easy one.

In the plane, and still more in higher space, this ceases at once to be the case; there is no single type of plane element which takes precisely the place of the linear segment, to the exclusion of all others, in fact the idea of form is one which first occurs in connection with space of more than one dimension, and is of fundamental importance in the classification and recognition of plane sets of points.