Sets of Points. Sequences. Limiting Points.

Given any set of numbers we have then corresponding to them a definite set of points on the straight line. Conversely a set of points on the straight line, or a linear set of points, is understood to mean any finite or infinite number of ‘points, such that their corresponding numbers form a finite or infinite set, for instance all the integral points, or all the rational points, or all the points of the segment (0, 1). This definition is to be regarded as equivalent to the following: a linear set of points consists of points of a straight line determined by a certain law which is such that (1) every point of the straight line either belongs to the set or does not, but not both, nor neither; (2) assuming that we are acquainted with all the characteristics of the points of the straight line, given any point we can determine whether or no it belongs to the set; and (3) having already obtained any number or collection of the points of the set, if there are any points of the set left, the law permits us to determine more.

A set which is contained entirely in another set is called a component of the latter set, and, if there are points of the latter set not belonging to the former set, it is said to be a proper component of the other; e.g. the Liouville points, which correspond to the Liouville numbers (Ch. I, § 7, p. 8), form a component of the set of all the irrational points.