Limit-points, Dense and Derived Assem blages.—A neighborhood or vicinity of a point p is a region small at will taken about p. If p be in space, a neighborhood may be a sphere hav ing p as centre; if p be in a plane or in a line, a neighborhood will naturally be a circle or a line-segment. The following discussion, con ducted for assemblages of points of a straight line, is readily extensible to other points assem blages. Denote by P any given assemblage of points of a line. If there be a point p, in P or not, such that in any neighborhood of p there is one point (and hence an infinity of points) of P, then p is a limit-point of P. If p be in P but not a limit-point, p is an isolated point of P. The assemblage of all the limit points of P is the first derived assemblage or derivative of P. The first derivative of is the second derivative of P, namely, ;and so on. If P be finite, its P"' contains no points, it is empty.. If P be infinite and.in a segment. P(') contains at least one point — a proposi tion of exceeding importance in function theory. If the nth derivative P® be empty and the pre ceding derivative contains one or more points, P is said to be of the first genus and nth If Pug/contain points for every positive integral value of n, P is said to be of second genus. Every point of a given derivative of P is a point of each preceding derivative, but P may contain points not in any of its derivatives. If some or all of the points of P are in an interval (a . (3) and if every sub-interval of the given one contains a point or points of P, P is said to be dense throughout the given interval. For example, the assemblage of points whose dis tances from a fixed point of the line are rational numbers is dense throughout every interval. If P be dense throughout a given interval, so is every derivative; in fact, each derivative in such case contains all points of the interval, and conversely. Hence one might lay down the definition: P is dense throughout an interval when and only when PO) contains every point of the interval. Obviously, if P is dense throughout an interval, P is of second genus, and so, too, are its derivatives. It follows that if P or one of its derivatives be of first genus, P is not dense in any interval. But it is not true that every P of second genus is dense throughout some interval.
Greatest Common Divisor, Least Com mon Multiple.— The equation P --"EQ will sig nify that the point assemblages P and Q are identical. Two assemblages having no element in common are said to be without connection. If P contains all and only the points of the as semblages Pa, . . every two of the latter being without connection, the fact is expressed by writing Pi, P,, . . .). A part of P is called a divisor of it, and P is a multiple of each of its divisors. The symbol, D(P1, is read greatest common divisor of P1, P1, and is the assemblage of their common points. M(P,, Ph . . ) is read least common multiple of P,, P,, . . . and is the assemblage of all the different points of the P's, it being understood that the latter have no common point. To ex press that P is empty one may write 2'). If and only if P and are without connection, D(P, Each dderivative of P is a divisor of every preceding derivative. If P is of sec
ond genus, then R), where Q is the assemblage of those points of P(9 that are not common to P(') , , , and R is the assem blage of those that are common.
Transfinite Derivatives.— R is therefore defined by the equation D (P(", . . ,) or by D (pm, Pe), .. . or by Ri-.7--D (POI), P(* where ... are a denumerably infinite assemblage of increasing positive in tegers. R is caned a derivative of P, but the order of the derivative is not expressible by a number of the sequence 1, 2, 3 . . ; these numbers are finite. The order of the derivative is denoted by the transfinite number 4), and one may write The first derivative of P('') is denoted by and the nth by If P(") have a derivative of transfinite order (0, it is denoted by POO. Continuation of the process yields P(40), P(0), • .), Pkw' , and so on endlessly. For any assemblage P of first genus, Puo--=`-0,an equation serving to characterize assemblage of first genus. Assemblages of the second genus are definable for which the derivative of any given transfinite order shall consist of a single specified point If D(P, P('))= 0, P is an assemblage of isolated points. From any assemblage P, an assemblage Q of isolated points is obtainable suppressing from P the assemblage D(P, P")), and one may write — D (P, P( ) . It is that if P be an assemblage of iso lated points, it is denumerable, though, as above noted, the converse is not true. Also, if Pe) is denumerable, so is P; but not conversely, for, for example, the assemblage of rational fractions is denumerable, while its first deriva tive is a continuum, namely, the assemblage of real numbers. Again, if P be of second genus, and if P(/4,a being finite or transfinite, be denumerable, so, too, is P denumerable. A very remarkable theorem is the following: if P be in any given interval and if be de numerable, the points of P can be enclosed in a finite number of sub-intervals having a sum less than any prescribed length Perfect Assemblages.— If and co incide P is called a perfect assemblage; in the contrary case, imperfect. For example, if P is the assemblage of points of the interval from pa to including fri and P is perfect; but if P includes only the points between pa and pa P is imperfect, for clearly includes pa andp. The definition just given is Cantor's. Another current definition is that by Jordan: P is per fect if it includes It has been proposed to distinguish the two by describing an assem blage, if perfect in Cantor's sense, as perfect, and, if perfect in Jordan's but not in Cantor's sense, as closed. It has been proved that if P be closed, the assemblage R which results on suppressing P(') from P is denumerable. But it is not true that every absolutely perfect assemblage is decomposable into a closed assemblage and a denumerable assemblage. The theory of perfect assemblages, though ex ceedingly subtle, is far simpler than that of imperfect assemblages. Every derivative of P is relatively perfect. There are absolutely per fect assemblages not dense in any interval.