A point a in every neighborhood of which there are points, distinct from a, of a set [x) is a limit point of [x] ; a either may may not be a point of the set [x]. If [x] is a linear set, a is either the greatest lower bound of the num bers of [x] greater than it or the least upper bound of the numbers of [x] smaller than it. A linear point-set without an upper bound is said to have the limit point + te and one without a lower bound to have the limit point — ce. The following theorem of Weierstrass is a simple corollary of the Heine-Borel theorem.
((Every infinite set of points lying on an in terval has at least one limit point)) If it had no limit point, then about every point of i there would be a segment including not more than one point of [x]. This set of segments, by the Heine-Borel theorem, would have a finite subset including all the points of i. But each interval of the subset not containin! more than one point of [x], jx] could contain at most a finite number of points, contrary to hypothesis.
If [x] is a linear point-set, and if we allow + oo — pus ao to figure as limit points, the theorem may, of course, be modified to read: Every infinite linear point-set has at least one limit point.
Theory of Linear Paint-sets.— The set of /knit points of a set [x] is called the first de rived set of [x]. The first derived set of the first derived set is the second derived set, etc. The theory of successive derived sets has been highly developed by the aid of the transfinite numbers of Cantor. (See ASSEMBLAGES). A set which includes its first derived set is said to be closed (abgeschlossen). A set included in its derived set is dense in itself (insichdicht). A set which is the same as its first derived set, i.e., which is both closed and dense in itself, is said to be perfect. An example of a set closed but not dense in itself is furnished by the num her 0, together with the numbers 1 —(n=1, 2, 3 . . .). A set dense in itself but not closed is the set of all rational numbers. The set of all numbers is an example of a perfect set.
A set whose derived set is the set of all real numbers on a certain interval is everywhere dense (iiberalldicht) on this interval. Such a set has a point between every two points of the interval. A set which is everywhere dense on no interval is nowhere dense. The first of the examples above is a nowhere-dense set, the second and third are everywhere-dense sets.
If [s.] is a dosed set, nowhere dense on an interval ab,. and c any point of ab not in [x] then there is a segment, including c, whose end points belong to [x], but which contains no point of [x]. Such a segment is called a point-free
segment of [x]. Thus, for every closed set [x] there exists one, and only one, set of point-free segments. No two of these segments overlap.
Therefore they are an enumerable(or denumer able) set (see ASSEMBLAGES), namely, there can —a be only a finite number of length > 2 — a _.... ' b—a finite number of length > — , and in general 4 b—a only a finite number of length > . The set [s] of end points and limit points of end points of the point-free segments is evidently a closed set and is identical with [x]. Since the set of point-free segments is enumerable, its end points are enumerable and thus the nowhere dense closed set is the derived set of an enumer able set of points together with the enumerable set itself. A consequence of this which we do not stop to prove here is that there is a one-to one correspondence between the continuum and any perfect subset of itself ; the argument de pends on the theory of point-free segments and the fact that the continuum is the derived set of the enumerable set of all rational numbers.' The notion, length of an interval, finds a generalization in the content of a point-set. [x] being any set of points on an interval i, let (Ti, (a• - , an be any finite set of intervals of length . . . , la, including every point of [x]. Let L=1,4 . . . ln. There are, of course, an infinity of numbers, L. Their greatest lower bound is called the content of [xl and written c[x]. If [x] is a closed set and S is the sum of the finite or infinite series obtained by taking the lengths of the point-free segments in the order of their magnitude, we have c[x]== 1—S, where 1 is the length of the interval i. Thus in the case of a dosed set [x] and its com plementary set [5], which consists of all points of the interval not points of [x], we have c[x]+c[ This, however, is not true of sets in general. For example, the set of all rational numbers between 0 and 1 has the con tent unity and the set of all irrational numbers on the same interval has the same content. The notion of length is, therefore, extended by the definition of content, only to closed sets and not to sets in general. A much more far reaching generalization of length has, however, been suggested by E. Borelu and carried out by H. Lebesgue' in connection with the theory of definite integrals (see below). (See also Measure and Measurable Assemblages, under ASSEMBLAGES, GENERAL THEORY OF).