CHAPTER XI - POTENCY OF PLANE SETS

Summary

The Theory of Potency in higher space is in all essentials identical with that in linear space, since, as has been shewn in Ch. VIII, all the points of a plane, or of space of any finite (or indeed countably infinite) number of dimensions, are of potency c, so that any set of points in the plane or higher space has the same potency as a certain linear set. Thus the only potencies which can occur are those which occur on the straight line, and, as there, the only known potencies are, beside finite numbers, that of countable sets a, and that of the continuum c.

Countable sets. A countable set is, as before, one such that the elements of it can be brought into (1, 1)-correspondence with the natural numbers. The coordinates of a countable set of points are, by Theorem 3, Ch. IV, clearly countable; conversely, any set of points whose coordinates are countable, is itself countable; thus the rational points in the plane or higher space are countable, and so are the algebraic points.

When arranged in countable order a countable set will be said to form a progression, precisely as on the straight line.

Cantor's Theorem, that a set of non-overlapping regions is always countable, has been proved in Ch. IX, as well as the theorem that a set of overlapping regions may be replaced by a countable number of them having the same internal points as the whole set.