BY
ERIC H. NEVILLE.
TRINITY COLLEGgp CAMBRIDGE.
Preface.
T h e origin of t h i s p a p e r w a s a d e s i r e f o r a d e f i n i t i o n of a p l a n e c u r v e w h i c h s h o u l d r e q u i r e a c u r v e t o be in s o m e sense in o n e p i e c e w i t h o u t r e q u i r i n g it t o be closed or t o b e of t h e v e r y s p e c i a l c h a r a c t e r of a J o r d a n c u r v e . T o t a k e a s i m p l e e x a m p l e , t h e r e m u s t b e s o m e s e n s e in w h i c h a l e m n i s c a t e d e p r i v e d of a n y p o i n t b u t t h e n o d e is a single c u r v e b u t a l e m n i s c a t e d e p r i v e d of t h e n o d e is t w o c u r v e s . T h e d i s c o v e r y of t h e p r o p e r t y of a s e t w h i c h I define in s e c t i o n 2o a n d d e s c r i b e b y s a y i n g t h a t a s e t is u n i t e d led t o t h e d e f i n i t i o n of a c u r v e o n a s u r f a c e w h i c h is g i v e n b e l o w in s e c t i o n 38 of t h e p a p e r , 1 b u t a r o u s e d a f r e s h d i s c o n t e n t , since t h i s d e f i n i t i o n g a v e n o clue t o t h e d i s t i n c t i o n in t h r e e - d i m e n s i o n a l s p a c e b e t w e e n a c u r v e a n d a s u r f a c e , a d i s t i n c t i o n w h i c h t h e d e f i n i t i o n since e v o l v e d , g i v e n in s e c t i o n 36, e n a b l e s m e t o d r a w .
A l t h o u g h t h e w o r k w a s b e g u n f o r t h e s a k e of a t h e o r y of d i m e n s i o n s , i t is n o t on a c c o u n t of t h e t h e o r y s u g g e s t e d in t h e c o n c l u d i n g s e c t i o n s t h a t t h i s p a p e r is p u b l i s h e d ; m u c h r e m a i n s t o be d o n e b e f o r e t h a t t h e o r y c a n b e p r o v e d v a l u a b l e o r v a l u e l e s s . B u t of t h e t h i r t y - n i n e s e c t i o n s of t h i s p a p e r t h e f i r s t t h i r t y - f i v e a r e c o n c e r n e d o n l y w i t h ideas w h i c h c e r t a i n l y h a v e t e c h n i c a l u s e as well as in- t r i n s i c i n t e r e s t ; a m o n g these, t h e f u n d a m e n t a l i d e a of w h i c h I h a v e f o u n d ~ n o 1 These definitions of a united set and of a curve on a surface were given in a short note entitled ~Definition of a plane curve~ in The Journal of the Indian Mathematical Society, Vol, vii. pp. ]75--I77 (~9x5), the set being there described as ~erfectly cwnne~ted.
(Added November I916) This statement only reveals my ignorance when it was written.
Undeniable traces are to be found in SCHOENFLIES' definition (Math. Annalen, bd. 58 (19o4), s. 21o) of a plane set of 'a special kind as eohe~'ent if every pair of its members can be joined by a simple path within the set, and in a footnote (American Journal of Mathe~tic#, v. 33 (19J])
t r a c e e l s e w h e r e is t h e idea a s s o c i a t e d h e r e w i t h t h e w o r d u n i t y , a n d t h e p a p e r is e s s e n t i a l l y t h e o f f e r of this idea f o r c o n s i d e r a t i o n .
T h e p a p e r c o n t a i n s n o proofs. I t seems to me t h a t p r o o f s a r e a l m o s t w o r t h - less unless g i v e n in f u n d a m e n t a l logical symbols, a n d a t p r e s e n t I h a v e n o t t h e t i m e t o p r e p a r e d e t a i l e d p r o o f s f o r p u b l i c a t i o n . M o r e o v e r , m a n y of t h e ideas h e r e p u t f o r w a r d , a n d in p a r t i c u l a r t h e idea of u n i t y a n d t h e d e r i v e d idea of a cell, are t o be j u s t i f i e d less b y p r o p o s i t i o n s t h a n b y examples, a n d in a n y case a n a c c o u n t of a s u b j e c t in g e n e r a l t e r m s is a v a l u a b l e p r e l i m i n a r y t o a f o r m a l d e v e l o p m e n t .
C e r t a i n a c k n o w l e d g m e n t s of d e b t I a m g l a d t o m a k e : m y d e f i n i t i o n of a c o n g r e g a t e of p o i n t s was s u g g e s t e d in p a r t b y JORDAn'S t r e a t m e n t of a closed set d'un seul tenant, in p a r t b y t h e desire for a d e f i n i t i o n e x p r e s s i b l e in a simple f o r m w i t h t h e s y m b o l i s m of a P r i n c i p i a Mathematica*), t h e use of this s y m b o l i s m a l o n e has e n a b l e d me t o c o n s t r u c t p r o o f s of t h e p r o p o s i t i o n s w h i c h I a s s e r t a n d of o t h e r s t h a t shew t h e i m p o r t a n c e of t h e v a r i o u s definitions, a n d to t h e influ- ence of RUSSELL a n d WHITEHEAD are d u e a m o n g o t h e r f e a t u r e s t h e f o r m in which t h e s u b j e c t m a t t e r of t h e p a p e r is d e s c r i b e d in section 2 a n d t h e r e c o g n i t i o n a t v a r i o u s p o i n t s of t h e r e l e v a n c e of t h e m u l t i p l i e a t i v e a x i o m .
T h i s w o r k was d o n e in t h e first i n s t a n c e in i g n o r a n c e of FR]iCHET'S t h e o r y 1 of d i m e n s i o n - t y p e s , b u t a c o m p a r i s o n of t h e d e f i n i t i o n s of section 36 with his d e f i n i t i o n s is to be f o u n d in t h e closing section. Since t h e idea of a u n i t e d set does n o t e n t e r i n t o FR~CHET'S t h e o r y , m y w o r k is a l t o g e t h e r i n d e p e n d e n t of his, b u t p r o b a b l y a n earlier a c q u a i n t a n c e with FR~CHET'S p a p e r w o u l d h a v e led m e t o d e v e l o p t h e p r e s e n t t h e o r y as a c o n t r i b u t i o n to t h e classification r a t h e r of a b s t r a c t a g g r e g a t e s t h a n of sets of p o i n t s . T h e u n i v e r s e with w h i c h I deal e x p l i c i t l y is less g e n e r a l t h a n FRkCH]~T'S class L (l. e., s. I45) a n d m o r e g e n e r a l t h a n his class E (1. c., s. i6o), i n a s m u c h as I a s s u m e t h e e x i s t e n c e of s e p a r a t i n g n u m b e r s b u t a s s u m e n e i t h e r t h a t t w o p o i n t s s e p a r a t e d b y t h e n u m b e r zero necessarily coincide n o r t h a t t h e i n e q u a l i t y x z ~ x y + yz is t r u e f o r every set of t h r e e e l e m e n t s ; e x a c t l y h o w m u c h of m y p a p e r is in f a c t i n d e p e n d e n t of t h e use of n u m b e r s I do n o t p. 319) in which LENNES remarks that obviously there are complete connected sets which contain no continuous arcs joining certain pairs of their points, the one writer using ~,simple paths, and the other ~continuous arc~ according to a precise definition. LE~ES' definition, unlike SCHOE~"
FLmS', is applicable to aggregates in general, but neither writer anticipates the kind of use made of the conception in the present paper.
1 MAURICE FafiCHET, ~Les dimensions d'un ensemble abstrait~, Math. Annalen, bd. lxviii, s. I4$
--i68 (i92o): the study of dimensions is now associated so closely with the name of BaouwEa that it is as well for me to state definitely that none of his work that I have seen indicates a perception of what I call unity, and that there is therefore nothing in this paper for which I am indebted to Prof. BROUWER.
know, b u t it is evident t h a t the definitions of united sets and of cells require only the assumption t h a t such limits as exist are members of the universe, a n d t h a t if limits are defined b y means of neighbourhoods, then plots a n d con/ined limiting points also can be defined, whether or n o t the definition of a neighbour- hood is numerical. On one detail, not of logic b u t of nomenclature, I incline to disagree with M. FR]~CHET, who wishes to restrict the phrase ,>of n dimensions~) to an aggregate whose t y p e is t h a t of the complete geometrical space with n coordinates; since there is no ordinal similarity between the types of dimensions and the signless real numbers, insistence on a one-one correlation in the parti- cular case in which the numbers are integral is actually misleading, a n d I prefer to allow t h a t the surface of a sphere m a y be described legitimately as well as popularly as two-dimensional. Nevertheless, not actually to contradict M. FR]~CHET, I have spoken of the dimension-integer rather t h a n of the number of dimensions of a set.
T a b l e o f Contents.
Preface.
Table of Contents.
1. Introduction.
2. Points and Separating Numbers.
3. Sets and Classes of Sets.
4. Bounded Sets and Sets with Separating Numbers Finite; the Span of a Set.
5. Units.
6. The Cardinal Number and the Reduced Number of a Set.
7. Reduction and Expansion.
8. Sequences.
9. Complements.
10. Neighbourhoods.
11. Limiting Points and Limited Sets.
12. Simple Sets.
13. Adherences and Coherences, Isolated Points and Internal Points, Edges and Boundaries.
14. Complete Sets and Closed Sets.
15. Extension and Completion.
16. Dense Sets and Perfect Sets.
17. Inseparability.
18. Congregates of Points.
19. Connected Sets.
20. United Sets.
91. The Separating Number of Two Sets.
22. Connex Sets.
23. The Relation between Connex Sets and Congregates.
.dvta mathcmatica. 42. Imprim6 le 12 Juin 1918. ls~s 9
24. Continuous Sets and Continua.
25. The Cells of a Set.
26. Sets Continuous in Every Part.
27. Plots.
28. Confined Limiting Points.
29. Brinks and Borders.
30. The Relation between Unity and Borders.
31. Partitions and Permeation.
32. Fronts.
33. Domains.
34. Capacious Sets.
35. Extreme Points.
36. The Dimension-Integer of a Set, and the Definitions of Curves and Surfaces.
37. Pure Sets and Mixed Sets.
38. Curves on Surfaces.
39. Dimension-Integers and Dimension-Types.
1. Introduction.
T h e v o c a b u l a r y of the t h e o r y of sets of points is evidence t h a t it has been left too much to the writers whose interests are primarily philosophical to insist on the e x t e n t to which t h e n a t u r e of the d e r i v a t i v e of a set of points F depends on t h e n a t u r e of t h e universe of points V of which F is a part. Technical mathe- maticians, instead of recognising f r a n k l y t h a t t h e y need t o deal sometimes with a universe of one kind, sometimes with a universe of a n o t h e r kind, h a v e expended a considerable a m o u n t of ingenuity in reducing their special universes b y a v a r i e t y of m o r e or less elaborate c o n v e n t i o n s to a common p a t t e r n . W h a t is most sur- prising is t h a t the chosen p a t t e r n is a universe of a different kind from Euclidean space a n d t h a t the c o n v e n t i o n a l reduction of Euclidian space to the s t a n d a r d f o r m is no easy task.
A universe of points V is said to be limited or closed if e v e r y infinite set of points in V has a ]imiting p o i n t in V. Thus the surface of a sphere is a closed universe, a n d so is the set formed of all the points inside or on t h e p e r i m e t e r of a plane triangle: on the o t h e r hand, t h e points inside a sphere do not compose a closed universe, n e i t h e r do the points inside a triangle, nor, to take the most i m p o r t a n t cases of all, do the points of Euclidean space or of a E u c l i d e a n plane.
Mathematicans, fully alive in some connections to the a d v a n t a g e s of dealing with open r a t h e r t h a n with closed sets, as is shewn in t h e use made of circular domains
in the theory of functions of a complex variable, have nevertheless shewn curious and I think unwarrantable reluctance to contemplate an open universe of points.
The first difficulty for which this reluctance is responsible in the treatment of Euclidean space is in the application of the words closed and open to unlimited sets, t h a t is, to sets tending in some manner to infinity. The obvious interpreta- tion, sanctioned b y the effects of establishing, for example, a point-to-point cor- respondence between the finite points of a plane and all the points except one on a sphere, is to regard an unlimited set as closed or open according as points at infinity have or have not an actual existence. This interpretation does not meet the difficulty: unless b y the invention of ideal points existence has been conferred on points at infinity, such limiting points as a set posesses must be in the finite p a r t of the universe, and a set which tends to infinity b u t has no finite limiting point simply has no limiting point whatever, and cannot be regarded as an open set without a change in the fundamental definitions; moreover, the introduction of ideal points and points at infinity is often accompanied b y a change in the definition of distance, and if a point at infinity is to be regarded as a limiting point the definition of a limiting point must be changed from the form which is naturally the first to be adopted.
We must not be supposed to assert t h a t the difficulties in this subject cannot be surmounted, or rather evaded, b y a multiplication of conventions. E v e r y s t u d e n t of the t h e o r y of functions of a complex variable knows that much can be accomplished b y introducing what is really an incomplete symbol zoo or oo, a symbol of which only the uses are defined, it being agreed for example t h a t to write
is to mean
[z--|
B u t even in this case the properties of oo as a limit are different from those of a n y other limit; for example, if (un) and (v~) are two sequences of complex num- bers with a common finite limit, u,~--v,~ must tend to zero, while if ( u ~ ) a n d (vn) both tend to oo, t h a t is, to infinity, u , ~ - - v n m a y behave in a n y manner as n in- creases, converging to a n y finite number, oscillating in any way, or diverging to infinity. I t is not, however, on technical troubles of a n y particular kind that the case for a candid consideration of facts is based: to be content rather to evade a difficulty than to analyse it and to understand it is the mark distinguishing not the technical mathematician from the philosopher b u t the computer from the mathematician.
2. Points and Separating Numbers.
The one requirement of the t h e o r y before us is a one-many relation, subject to certain conditions, between signless real numbers and cardinal couples of indi- viduals. The individuals, in virtue solely of the fact t h a t every member of the converse domain of the fundamental relation is a pair of them, are called points, a n d the real number associated with a pair of points is called the separating n u m b e r of the two points, or between the two points, forming the pair, or the n u m b e r separating one of the points from the other. We use for points the letters v, w, x, y, z a n d we denote the separating number of x and y b y x y ; because the separating number of x and y is a number related to the cardinal couple composed of x a n d y, not to either of the ordinal couples composed of the same individuals, y x is the same as x y , and the further hypotheses as to the n a t u r e of the fundamental relation m u s t be enumerated. I t is assumed not only t h a t it is between pairs of points t h a t there are separating numbers, b u t also t h a t if there are separating numbers y v and z w there is a separating number y z , an assumption we express b y saying t h a t our points compose a universe. I t is assumed t h a t if the two points forming a cardinal couple are identical, the number associated with the couple is zero, t h a t is, t h a t if x is a n y point, the separating number x x is zero, b u t it is n o t assumed t h a t the separating number of two distinguishable points is necessarily different from zero. I t is assumed t h a t the class of signless real numbers defined by the criterion t h a t ~ belongs to the class if for every group of three points x, y, z the separating number x z is equal to or less t h a n the sum of the separating numbers x y , y z provided only t h a t x y and y z are both less t h a n Q is a class which has members other t h a n zero: from the form of the class, if a belongs to it so does every number less t h a n a; in Euclidean space every real number belongs to this class, a n d by dealing with the class itself we avoid both the indefinite and unnecessary restriction involved in choosing one of its members arjd the introduction symbolically of an infinite number. F r o m the assumption just explained it follows t h a t if the separating number y z is zero, then for every point x for which either x y or x z is sufficiently small, x y is equal to x z , and we make the final assumption t h a t in fact if y z is zero then for every point x the numbers x y and x z are the same.
I t m u s t be realised t h a t the separating number of two points need not be t h e distance between the points in a n y space in which t h e y can be represented.
As an example of the latitude we allow, the universe might consist of the sur- faces of two non-intersecting spheres in Euclidean space, a n d we might take for t h e separating number of two points on the same sphere their least geodesic
distance a p a r t and for the separating n u m b e r of two points of which one is on each of the spheres their distance a p a r t in space. Moreover, the separating n u m b e r regarded as a function of the two points on which it depends, m a y v a r y consi- derably in form without involving a n y change in the classification of classes of points with reference to a n y of the properties which it is our object to describe.
Indeed, in the case in which the universe is space of a finite number n of dimen- sions and a point x is determined b y its n coordinates xl, x~ .... x,, while some writers use for the separating n u m b e r of two points x, ~ the distance between them, which if the axes are orthogonal is I/I {Z( x ~ - ~?r )~}, others, following JORDAn, use for the separating number ~ [ x r - - ~ [ , and others again, including MOSKOWSXI, use ~ the greatest among the numbers of the form [ x ~ - - ~ [ ; none of the proper- ties of classes of points which separating numbers are used to described are af- fected b y a choice between these three functions.
An excellent example of the mode of embodying ideas in the definition of separating numbers when these numbers are divorced entirely from distances is to be found in the geometrical t r e a t m e n t of the complex variable. To j u s t i f y the usual method of dealing with the point at infinity, it is common to suppose a sphere drawn to touch the plane of the variable z at the origin
O,
and to say t h a t often a value z of the complex variable is a s s o c i a t e d not with a point zp of the plane b u t with a corresponding point z, of the sphere, z, being obtained b y joining zp to the point of the sphere diametrically opposite to O; if we wish to concentrate our attention on the plane, we have only to say t h a t limiting points are determined not with reference to distances in the plane b u t with reference to a distinct system of separating numbers, the separating number of two points ~p, ~p of the plane being the distance between the corresponding points~,, ~, of the sphere or some other n u m b e r suggested b y geometry, b u t the most satisfactory method is to regard the complex variable itself as a point, and the separating number not as a n u m b e r separating two points of a plane or of a sphere b u t as a number separating two values of the complex variable. P r o b a b l y the most useful n u m b e r separating the complex numbers 4, ~ is
which is the actual distance in space between the corresponding points on a sphere of unit diameter, b u t the n u m b e r
~ T h i s last c h o i c e h a s t h e a d v a n t a g e of b e i n g f a r m o r e w i d e l y a p p l i c a b l e i n s p a c e of a n i n f i n i t y of d i m e n s i o n s t h a n e i t h e r t h e C a r t e s i a n m e a s u r e or JORDAN'S: I h a v e f o u n d t h e con- s i d e r a t i o n of t h i s k i n d of space w i t h t h i s species of s e p a r a t i n g n u m b e r v a l u a b l e in r e m o v i n g p r e j u d i c e s .
+ + Ixl + lyl) O + + li/I)},
constructed on JORDAN'S model, is equally effective; denoting either of these separating numbers by ($s we can define the uses of the symbol ~ b y the assertion t h a t ( z ~ ) denotes I / V ( I + [zl ~) or i / ( i + ]xl +]y[)~ a n d t h a t ( ~ o o ) is zero, and oo t h e n exists, for the s t a t e m e n t t h a t the complex number oo exists means only t h a t the numbers separating oo from all complex numbers including itself are defined and satisfy certain conditions.
More interesting is the t r e a t m e n t of R I E M A ~ surfaces, which we m a y illustrate by the simple example of the surfaces connected with the representation of w'/2.
In the primitive form, these are a double plane and a double sphere, with cross- connections along the positive halves of the real axes in one case and along overlying semicircles in the other case. Logically, in the space in which w ~/2 is a one-valued function each point consists of a complex number z and a variable t which has only two values (most conveniently, I a n d 2); if m and n coincide and if in a plane corresponding to the one variable z the chord joining ~p to ~p does not cross the positive half of the real axis, or if m and n are different and if in this plane the chord does cross this half-axis, then the number ~m~, separat- ing the RIEMA~r point Sm from the RIEMA~N point ~, is a n u m b e r ( ~ ) s u c h as was defined in the last paragraph, while in other cases the separating number
~ , ~ is the smaller of the two numbers (zo) + (~o), ($ ~ ) + (~ ~o ), where o denotes the complex n u m b e r zero. Here we have distinct points at zero distance apart, for it is simpler to say t h a t there are two points o~, 02 a n d two points oo~, oo 2 and t h a t o, o2 and ~ ~ are zero t h a n to say t h a t while almost every point of the RIEMAN3 surface consists of a complex number a n d a two-valued variable there are certain points which consist of complex numbers alone.
3. Sets and Classes of Sets.
Our subject is the classification of classes of points, and for the sake of emphasis a class of points is called a set of points, or simply a set, while set is n o t regarded in a n y other sense as equivalent to class. A set is usually deter- mined by the s t a t e m e n t of a group of conditions which a point is to satisfy if it is to belong to the set; if there are no points satisfying simultaneously all the prescribed conditions, the group of conditions is said to determine the null set.
For sets in general we use the letters F , J , O, O, ~2, the universe is itself a set, which is denoted by V, and the null set is denoted by .4. F o r a general class of sets we use 7, while u ' F a n d ~'.F denote classes of sets related in a special
m a n n e r to a set F; thus each of the letters x, 9 m a y be regarded as denoting either a relation between a class of sets and a single set or an operation b y which from a n y set is derived a class of sets having a particular relation to t h a t set. Similarly we use L a t i n capitals other t h a n F to denote particular rela- tions between one set a n d another, or operators deriving from a n y given set certain other sets related to it, while we use K ' ( x , F ) and T ' ( x , F ) for special sets connected with a point x and a set F.
When we have to deal with classes of sets, the whole t h e o r y of selections is applicable to our work, a n d we have to consider whether or n o t in a class of sets ~ every member F has a representative. We say t h a t the universe is a ZERMELO universe if in every class of existent sets each set has a representative, or in other words if every class of m u t u a l l y exclusive existent sets is multipliable, and we say t h a t a set F is a ZERMELO set if every class of m u t u a l l y exclusive existent sets contained in F is multipliable; in a ZERMELO universe every set is
a ZERMELO set.
4. Bounded Sets and Sets with Separating Numbers Finite;
the Span of a Set.
Sets and universes m a y be classified according to the m a g n i t u d e of the separating numbers which are possible between points belonging to them. A set is bounded with reference to an origin v if there is a finite upper limit or m a x i m u m to the numbers separating its points from v. If there is a number a, necessarily different from zero, such t h a t if x, y belong to F , then x y is less t h a n a, there is a least number q which is such t h a t if x , y belong to F and a is greater t h a n q, then xy is less t h a n q; this n u m b e r is an upper limit or maxi- mum to numbers separating pairs of members of F, and is called the span of F . If x z is never greater t h a n x y + y z , a set bounded with reference to a n y origin has a finite span.
A set is a set with infinite separating numbers or a set with separating numbers finite according as it does or does not contain two points whose separat- ing number is infinite. E v e r y set with separating numbers limited is a set with separating numbers finite, b u t the converse is n o t true: for example, in Euclidean space, not subjected to a n y conventional closing, the distance between a n y two points is finite, b u t there is no upper limit or m a x i m u m to the distances possible;
by replacing the points and lines of Euclidean space by ideal points a n d lines we can obtain a space in which there are actual points a t infinity, and in this space infinite distances are possible.
5. Units.
A set which includes a point x need not include all the points separated from x by the number zero, and we denote by T ' ( x , / ' ) the set composed of all the points of F separated from the point x of /~ by the number zero, calling T' (x,l") the unit of F containing x; if x does not belong to F, the set T ' ( x , F ) is defined to be null, even if I" contains points separated from x b y the n u m b e r zero. A set is called a unit of r if it is the unit containing some point of /~, a n d we denote b y v'I" the class of units of F . If y is a member of T ' ( x , F ) , the units T' (x, F) and T' (y, F) are identical, and x a n d y contribute the same member to the class v' F.
If F is n o t null, the class v ' F is a class of m u t u a l l y exclusive existent sets whose sum is / ' ; even if F is null, the null set is n o t a member of v ' F : if F is null, ~'F is the null class of sets, n o t the class whose member is the null set.
The set T' (x, F) is a unit of t h e universe, and while every u n i t of a set F is contained in some unit of the universe, a unit of /~ need not coincide with the u n i t of F which contains it, and the class v ' F need not be contained in the class v' F.
6. T h e Cardinal N u m b e r and t h e Reduced Number o f a Set.
The number of different points belonging to a set F is called the cardinal number of the set and denoted b y N c ' F . A set is called finite or infinite accord- ing as its cardinal number is or is not inductive, and is called singular if it has only one member, plural if it has more members t h a n one.
For m a n y purposes the n u m b e r connected with a set F which is of greatest importance is n o t the number of points belonging to F b u t the n u m b e r of units contained in F, t h a t is, the n u m b e r of sets belonging to ~ ' F ; this number, Nc'~'F, we call the reduced n u m b e r of F and denote b y N c r ' F . We say t h a t a set is scattered finitely or infinitely according as its reduced number is or is not induc- tive, and we call a set a u n i t set if it has only one unit, a multiple set if it has more units t h a n one.
I n formal work unit sets, both singular and plural, require a t t e n t i o n o u t of all proportion to their interest, which is negligible. The universe m a y itself be finite, in which ease every set is finite, or finitely scattered, when every set is finitely scattered, but none of our definitions give a n y b u t the most trivial results when applied to sets in a finitely scattered universe.
7. Reduction and Expansion.
To reduce a set is to omit from it a n y set of points each of which is separated by the n u m b e r zero from some one of the points retained. Reduction is possible as long as the set contains units which are not singular, a n d we say t h a t a set or a universe is fully reduced if all its units are singular, t h a t is, if it includes no two distinguishable points whose separating n u m b e r is zero. If a set is fully reduced its cardinal number is the same as its reduced number, b u t the converse implication holds only if the set is finite. A set is called a reduced form of a set F if it is contained in F a n d includes one and only one member of each unit of F ; in more technical language, the reduced forms of F are the selected sets of the class of units of F. To say t h a t reduced forms of F exist is to assert t h a t the class of units of F is multipliable, a n d since this assertion can not be made of every set unless the multiplicative axiom is assumed, we content ourselves with describing a set as reducible if there are reduced forms of it. In a fully reduced universe every set is fully reduced, but a reducible universe m a y contain sets t h a t are not reducible; in a ZERMELO universe all sets are reducible, b u t we have no reason to suppose t h a t if every set is reducible the universe is a ZERMELO universe, for every class of unit sets m i g h t be multipliable while some classes including multiple sets were not multipliable. I t is i m p o r t a n t to notice t h a t every finitely scattered set is reducible.
The converse of reducing a set is adding to it a n y set each of whose points is separated by the number zero from some point already included, and this process is called expanding the set. Expansion is possible if there are units of the set which do n o t coincide with the units of the universe containing them, a n d we say t h a t a set is fully expanded if every pair of points of which one belongs to the set a n d the other does n o t has a separating number different from zero.
The set obtained b y expanding F fully is the sum of the class of units of the universe containing points of F , and we denote it by E'F. If F is a set having points in common with a n o t h e r set J , to e x p a n d F in d is to add to F a n y members of d separated b y the n u m b e r zero from points belonging to both F a n d J , a n d we denote b y E 4 ' F the set obtained by expanding as far as possible in z/ the p a r t of F contained in ,d; the set E,~'F is contained in both E ' F and d , b u t there m a y be points common to E ' F a n d z / w h i c h do n o t belong to Ea' F .
8. Sequences.
Of infinitely scattered sets the simplest are fully reduced sets whose members can be p u t into a one-one correlation with the inductive cardinals; if such a
.Aeta m a t h e m a t / c a . 42. Imprim6 le 12 juin 1918. 1828s l 0
correlation has been established, a n d xn is the point corresponding to n, then the relation of n to x~ is called a sequence and denoted b y (x~), a n d the set of points is the converse domain of this relation.
9. Complements.
The points which do n o t belong to a set /1 compose a set called the com- plement of F , which we denote by C' F . If F a n d J are a n y two sets, the points of z/ which do not belong to F form a set called the complement of /~ in J , which we denote by
C~'F;
if J is contained in F, there is no point of ~ / w h i c h does n o t belong to F , a n dC,~'F
is null.10. Neighbourhoods.
B y the neighbourhood of x with radius Q we mean the set composed of all points separated from x b y numbers less than Q. The character of a neighbour- hood depends on the n a t u r e of the universe; for example, if the numbers separat- ing x from points outside the unit to which it belongs have a lower limit or minimum ~ other t h a n o, then for a n y value of ~ between o and a the neighbour- hood of x with radius Q coincides with the unit which includes x. W h a t e v e r the character of the universe, the neighbourhood of a n y point x with zero radius is null, but every other neighbourhood of x includes at least the one point x, and neighbourhoods with radii t h a t are not zero m a y be distinguished as existent neighbourhoods.
11. L i m i t i n g P o i n t s and L i m i t e d Sets.
H a v i n g agreed r a t h e r to classify universes according to their properties t h a n to consider only universes of special kinds, we define the limiting points of a set by the definition t h a t x is a limiting point of F if there is a point of F outside the u n i t containing x in e v e r y existent neighbourhood of x, and we a d m i t no modification which leads to a change in the content of the set of limiting points of a n y set whatever. The set whose members are the limiting points of /~ is called the derivative of F a n d denoted by D ' / ' .
I t is a f u n d a m e n t a l proposition t h a t every set which has a limiting point is infinitely scattered; the converse is not true. There are universes in which every infinitely scattered set has a limiting point, and such universes are said to be limited; for m a n y purposes the distinction between a limited universe a n d an unlimited universe is the most i m p o r t a n t distinction between one universe
a n d another. In a n y universe a set is called unlimited or limited according as it does or does n o t contain an infinitely scattered set w i t h o u t a limiting point, and a limited universe m a y be defined as a universe in which every set is limited.
I t was proved by WEIERSTRASS a n d is said by Y o u n g to have been known to earlier writers t h a t in the simplest types of space every infinitely scattered bounded set has a limiting point. B u t it is easy to show t h a t this proposition is not true for all spaces: if the universe consists of all signless rational numbers in a finite stretch and separating numbers are arithmetical differences, then every set is bounded b u t no set is limited which in a wider universe would have irra- tional limits; again, if the point is a progression of numbers (xl, z2 . . . ) a n d the separating n u m b e r ~ is the greatest among the differences I ~1 - - x~ I, I ~l - - x2 I, - . . , t h e sequence (i, o, o . . . . ), (o, i, o . . . . ), (o, o, I . . . . ), . . . . in which the n t h point has its n t h coordinate u n i t y and its other coordinates zero, is an infinitely scat- tered bounded set, with span unity, b u t with no limiting points.
12. Simple Sets.
A set which is the converse domain of a sequence m a y have no limiting points, or a n y finite n u m b e r of limiting points; indeed, as is shewn by the familiar correlations of the rational numbers in a finite interval with the natural numbers, the cardinal number of the derivative of such a set m a y be an infinite n u m b e r greater t h a n the cardinal n u m b e r of the set itself. A set is said to be simple if it is the converse domain of a sequence and has n o t more than one limiting point, a n d a sequence is said to be simple if its converse domain is simple. The use of simple sets comes from the theorem t h a t each limiting point of a n y ZERMELO set F is the unique limiting point of some simple set contained in F , from which it follows t h a t a ZERMELO universe is limited if every simple set has a limiting point.
In Euclidean space of a n y number of dimensions, if (yn)is a sequence whose converse domain is simple a n d unlimited and x is a n y point of the universe, a n d if the separating number of two points is the distance between them, t h e n xy~ tends to infinity, b u t this property is not common to all unlimited universes.
13. A d h e r e n c e s and Coherences, I s o l a t e d P o i n t s and I n t e r n a l P o i n t s , Edges and Boundaries.
There is no general relation of inclusion between a set a n d its derivative.
Points which belong to /~ b u t n o t to D ' r are called isolated points of /~ and
compose a set known as the adherence of F , and points which belong b o t h to F a n d to D ' F form a set called the coherence of F. Points of F which are limiting points of the complement of F constitute the edge of F , a n d points of F which do n o t belong to the edge of F are called internal points of F. Of the points of the derivative D' F, those which belong to F can of course be distinguished from those which belong to C'F, b u t it is unnecessary to invent names for the two sets indicated by this distinction, for the points common to D ' F a n d F compose the coherence of F, a n d the points common to D ' F and C ' F form the edge of the complement of F . Hero we m a y add t h a t the sum of the edges of F and C ' F is called the b o u n d a r y of F .
If x is an isolated point of / ' , every point of the unit of F containing x is an isolated point, a n d the unit itself is called an isolated unit of F . If x is a limiting point of F , every point separated from x by the number zero is a limi- ting point of F , and therefore every derivative is a fully expanded set.
14. Complete Sets and Closed Sets.
We say t h a t a set of points, limited or unlimited, is complete if it contains all its limiting points; thus in Euclidean space hyperbolas and helices are no less complete t h a n circles, and a universe is necessarily complete. I t is only to sets t h a t are both complete and limited t h a t we give the description of closed; a set is open if it is either incomplete or unlimited. Since a universe can not be in- complete, to say t h a t a universe is closed is t h e same as to say t h a t it is limited;
moreover, in a limited universe all sets are limited, a n d therefore in a limited universe there is no difference between closed sets and complete sets.
15. Extension and Completion.
Extension of a set is addition to it of a n y set of its limiting points which it does not include originally, t h a t is, addition of a n y part of the edge of the complement. If to a set F is added the whole of the edge of C ' F , the set F is said to be completed or fully e x t e n d e d , and we denote the set obtained, which m a y be described simply as the sum of F and D' F, by G' F. If F is not limited, neither is G'F, a n d therefore G'F is n o t necessarily closed, b u t the limiting points of G ' F are those of F itself and belong to G'F, whence it follows t h a t G ' F is in all circumstances complete. The adherence of G'F is the same as the adherence of F , and so G'F need n o t be fully expanded, a n d for some pur- poses we have to use the set E' G'F obtained by expanding G ' F fully, which is the same as the set G ' E ' F obtained by completing E ' F .
16. Dense Sets and Perfect Sets.
If e v e r y p o i n t of a set is a limiting point of the set, the set is said to be dense, and a set which is b o t h complete and dense is called a perfect set, whether or n o t it is limited. A dense universe is necessarily perfect, b u t although the commonest and most i m p o r t a n t universes are dense, there is no logical necessity for a universe to be dense even if it is infinitely scattered. The simplicity of a dense universe comes chiefly from the fact that in a dense universe if F is any set w h a t e v e r e v e r y point is a limiting point either of F or of C ' F ,
Any set obtained b y extending a dense set is itself dense, and therefore the set obtained b y completing any dense set is perfect. Although a dense uni- verse must contain perfect sets, it is not every dense universe t h a t contains sets both perfect and limited.
17. Inseparability.
We say t h a t two sets are inseparable if the corresponding completed and fully expanded sets have at least one point in common; thus F and d are inse- parable if there is a point which belongs to both E ' G ' F and E ' G ' d , that is, if there is a unit of the universe which contains members of both G ' F and G ' J . With this definition the null set is not inseparable from any set.
18. Congregates of Points.
A set in any universe we call a congregate of points if in e v e r y expression of it as the sum of two existent sets the two components are inseparable. If two sets have a common point they are certainly inseparable, and therefore although it is only a division into mutually exclusive sets t h a t it is natural to contemp- late when considering whether or not a s e t is a congregate, there is no need to complicate the definition of a congregate b y requiring the sets considered to be mutually exclusive; since, however, the null set is not inseparable from any other set, it is necessary to insist t h a t in e v e r y case neither of the components is null. Removing the word inseparable we m a y say t h a t a set is a congregate if however it is expressed as the sum of two existent components the completed and fully expanded sets corresponding to the two components have at ]east one common point.
19. Connected Sets.
American writers have a d o p t e d a definition of considerable interest, which implies cohesion of a higher o r d e r t h a n is essential to a congregate: a set is said to be connected if in e v e r y expression of it as the sum of two existent comple- m e n t a r y c o m p o n e n t s one of these c o m p o n e n t s includes a limit of the other; t h e definition recalls the Dedekindian axiom of continuity. Obviously e v e r y connected set is a congregate, and it is easy to shew t h a t e v e r y complete congregate is connected, b u t t h e set formed of those points of a sphere which do n o t lie on a p a r t i c u l a r great circle presents a simple example of a congregate t h a t is n o t connected.
T h e step from a congregate to a c o n n e c t e d set introduces precisely the con- dition essential for a t h e o r e m t h a t is b o t h interesting and valuable: a connected set t h a t includes b o t h a p o i n t t h a t belongs t o a set ,I/ and a p o i n t t h a t does n o t belong to J necessarily includes a p o i n t on the b o u n d a r y of d .
20. U n i t e d Sets.
We come now to an idea 1 which appears to be of f u n d a m e n t a l importance.
We say t h a t a set F is u n i t e d if e v e r y pair of members of /" is c o n t a i n e d in a closed congregate contained in the f u l l y e x p a n d e d set E ' F . I t can be shewn t h a t e v e r y closed congregate is united, b u t while a united set is a congregate in the original sense it m a y be incomplete or unlimited and m a y indeed be b o t h incomplete and unlimited. Closed congregates are necessary as a means to t h e definition and s t u d y of united sets, b u t the p r o p e r t i e s of closed congregates which are valuable a n d arise from the combination of t h e i r qualities seem to be preci- sely those which belong equally to all united sets.
As simple an example as there is of an open united set is the set f o r m e d of all t h e points on one side of a straight line in a r e d u c e d Euclidean plane, t h e distance between two points being chosen for t h e i r separating number. I n Eucli- dean space of a n y n u m b e r of dimensions, if y a n d z are a n y two points t h e set formed of y a n d z and all points between t h e m on t h e straight line t h r o u g h t h e m is called the closed chord joining t h e m ; this set is closed a n d connected. If y a n d z lie on t h e same side of a n y straight line in a plane e v e r y p o i n t of the closed c h o r d joining t h e m lies on t h e same side of t h a t line. H e n c e in t h e set described, the closed c h o r d joining a n y two points of t h e set is a closed eongre- 1 This idea seems narrowly to have escaped formulation by Prof. R. B/~IIcE, but I have failed to find an account of it in any of his writings that I have seen.
gate containing the points and contained in the set, and the set is therefore united, although it is unlimited because it extends to infinity and incomplete be- cause the points of the straight line used in defining it are limiting points which it does not include. I n Euclidean space a set F is called convex when if g and z are a n y two points belonging to r every point of the closed chord joining y a n d z belongs to F, and after w h a t has been said it is h a r d l y necessary to point o u t t h a t if the separating numbers are distances every convex set is united.
The n a t u r e of u n i t y m a y be illustrated f u r t h e r by means of a reduced Euclidean circle deprived of one point or of two points, the separating numbers again being distances. If F is such a circle and v, w are distinct points of F, then F itself, the set obtained b y removing v from F, and the set obtained by removing from F both v a n d w, are all limited congregates, b u t only the first of them is complete.. The second of these three sets is, however, united, since if y, z are a n y two of its points the arc of r which has y and z for its end points a n d does not include v is a closed congregate containing y and z and con- tained in the set in question. B u t the third set is n o t united, for there are two arcs of /~ which have v a n d w for end points, a n d if y is a n y point distinct from v a n d w in one of these arcs a n d z is a n y point distinct from v and w in the other, every complete congregate contained in /~ a n d containing both y and z includes either v or w, a n d there is therefore no closed congregate containing y a n d z and contained in the set obtained by depriving F of v and w.
An example even more instructive t h a n the last is given by a lemniscate or a n y other figure which might be described as a figure-of-eight. This is a closed connected figure a n d the set obtained by removing from it a n y one point is a congregate although it is incomplete; if the point removed is a n y other point t h a n the node, or if the curve is supposed to have two distinct points a t the node and only one of these is removed, the set remaining is united, b u t if the node is regarded as one point a n d is removed, or if all the points a t zero distance from the node are removed, there remain two distinct sets, each in itself united, which do not together form a united whole.
A multiple united set is necessarily connected, b u t an example proves t h a t the converse is not true, and t h a t even a complete connected set m a y fail to be united. As simple an example as a n y is given b y a branch of a hyperbola in a Euclidean plane together with a n y stretch contained in one of the asymptotes and a sequence of lines in which every line is parallel to this a s y m p t o t e and cuts the curve and the sequence has the a s y m p t o t e for sole limit: this set is connected a n d is complete if the stretch consists of the whole a s y m p t o t e b u t every conneeted component which includes both a point in the a s y m p t o t e and a
point not on t h a t line is unbounded and is therefore open since the space is Euclidean. In this example it is to be remarked t h a t the set deprived of the points belonging to the a s y m p t o t e is a united set, whence follows the interesting theorem t h a t extension, whether partial or full, of a united set m a y result in a loss of the unity.
Since every multiple u n i t e d set is connected, such a set cannot leave a n y other set w i t h o u t crossing the b o u n d a r y of t h a t set. In a dense universe, if a unit contains b o t h points of a set F and points of the complement C' F, either all the points of F in the u n i t or all the points of C ' F in the unit belong to the b o u n d a r y of F, a n d therefore in a dense universe the theorem just stated can be enunciated of all u n i t e d sets and n o t of such only as are multiple. More generally, if a united set which is n o t contained in an isolated u n i t of the uni- verse includes both a point of a set F and a point of the complement of F , it includes a point of the b o u n d a r y of F.
To illustrate the way in which the result enunciated in the last p a r a g r a p h m a y be used to endorse the conclusions of common sense, let t h e universe be a reduced Euclidean plane, let g be a straight line in this plane, a n d let O be the set formed of points on one side of this straight line a n d ~ the set formed of points on the other side, separating numbers being distances. Then as a l r e a d y remarked, O and 9 are both united. If, however, y is a point of O and z a point of q), then since z belongs to the complement of O and every closed congregate is connected, every closed congregate containing both y a n d z contains a point of t h e b o u n d a r y of O, t h a t is, a point of d . If t h e n F is the sum of O a n d O , there are points y and z belonging to F such t h a t every closed congregate con- taining y a n d z contains a point of the complement of F, or in other words such t h a t there does not exist a closed congregate containing y a n d z a n d contained in F ; t h a t is to say, the complement of a straight line in a Euclidean plane is the sum of two united sets b u t is not itself united. If, however, to the sot F we a d d a single point x belonging to the line g , t h e n if y and z are a n y two points of the resulting set, the sum of the closed chords joining y to x and x to z is a closed congregate containing y and z a n d contained in the set considered, a n d this set is therefore united.
21. The Separating Number of Two Sets.
From this point we propose a digression to compare the qualities we indi- cate by calling a set a congregate with the properties associated with the word
connex. The digression, interesting in itself, begins with the i n t r o d u c t i o n of a notion t h a t can be made to serve m a n y useful purposes.
If F and , / arc a n y two sets, the class of signless real numbers each of which is not less t h a a the number separating some point of F from some point of J is such t h a t if a belongs to it so does every n u m b e r greater t h a n a; this class has therefore a lower limit or minimum, a n d this lower limit or minimum we call the separating number of the two sets F a n d J . If the separating num- ber is a true minimum, t h a t is, an a t t a i n e d minimum, of the class b y which it is defined, t h e n there exist points y, z belonging one to each of the sets F, ,/, such t h a t the separating number of the two sets is the separating n u m b e r y z of the two points, b u t if the separating number of the two sets is an u n a t t a i n e d lower limit, then even to be entitled to assert t h a t either there are a point y belonging to one of the sets and a simple sequence (zn) whose converse domain is contained in the other set such t h a t the separating number y z , , tends to the separating n u m b e r of the sets, or there are simple sequences (yn), (zn) whose converse domains are contained one in each of the sets F, , / such t h a t y n z n tends to the separating number of the sets, we m u s t know t h a t the sets are Zer- melo sets. I t is convenient to modify the conclusion just reached as to Zermelo sets by using the fact t h a t a simple sequence (xn) such t h a t for every value of n t h e point xn belongs to a set 0 is either limited or unlimited and if limited has a limiting point belonging to D' q}. We deduce t h a t if two sets F, ,~ are Zermelo sets, a n d their separating n u m b e r is r then either there are points y , z belonging one to each of the completed sets G ' F , G ' , / such t h a t the separating n u m b e r y z is equal to r or there are a point y belonging to the set obtained by completing one of the sets F, , / a n d a sequence (zn) whose converse domain is contained in the other of the sets F , , 4 and has no limiting point, such t h a t y z,~ tends to r or there are sequences (y~), (z~) whose converse domains are con- tained one in each of the sets and have no limiting points, such t h a t ynz,, tends to r This is a result to which it is easy to apply knowledge of distinctive pro- perties of the sets F , J or of the universe; for example, in Euclidean space of a n y finite n u m b e r of dimensions with distances for separating numbers the se- cond case cannot occur, while in a n y limited ZER~ELO universe if the separating n u m b e r of two sets is Q, there are points y, z belonging one to each of the cor- responding completed sets such t h a t y z is equal to r I t is n o t e w o r t h y t h a t extension and expansion, partial or full, are w i t h o u t effect on t h e separating n u m b e r of two sets.
An example in which the separating n u m b e r of two sets is an u n a t t a i n e d limit will prove useful for reference. In a Euclidean plane with distances for
A c t o m ~ h e m a t i c a . 42. Imprim6 lo 12 Juin 1918. ~lms 1 1
separating numbers, let F be a branch of a hyperbola and d a line parallel to one of the asymptotes of the curve, a t distance Q from this asymptote, and on such a side of t h e a s y m p t o t e if Q is n o t zero t h a t it does not intersect F. Then the separating n u m b e r of F a n d J is ~, b u t F a n d ~ / a r e both complete a n d the distance of a n y point of F from a n y point of J is greater t h a n e. This example emphasises the difficulty of giving a satisfactory description of such features as this b y a process beginning with a conventional closing of the universe; there are different methods of closing the Euclidean plane, b u t in all of t h e m parallel lines meet at infinity, a n d so it m u s t be agreed if the universe is closed either t h a t in this example the separating n u m b e r of F a n d J is zero whatever t h e value of Q, on t h e ground t h a t r a n d , / have a common point, or t h a t a point at infinity m a y be a t a distance greater t h a n zero from itself.
B y means of t h e general idea of the separating number of two sets it is possible to simplify particular definitions in m a n y i m p o r t a n t cases. F o r example, if J has only one member, x, the separating n u m b e r of J a n d F is called t h e n u m b e r separating x from F , a n d if this number is zero either there is a point o f / ~ in the u n i t of the universe which contains x or x is a limiting point of r . Again, the number separating x from the set obtained b y depriving F of all its points in the u n i t of the universe containing x is the isolating radius of x a n d F ; if y is a n y point separated from x b y a n u m b e r which is n o t zero b u t is less t h a n this isolating radius, y belongs to the complement of /1, a n d if x belongs to F , the unit o f / " containing x is an isolated u n i t if the isolating radius is n o t zero but x belongs to the coherence of r if the isolating radius is zero.
22. Connex Sets.
The first definition of a connex set was given by CA~TOR, according to whom a set F is eonnex if given a n y signless number r other t h a n zero and a n y two points y, z of F, there can be formed a set d including y a n d z, contained in F , a n d consisting of a finite number n of points which can be correlated with t h e first n natural numbers in such a way t h a t if xm corresponds to m, the first point xl is y, the last point x~ is z, a n d every pair of consecutive points of , / has a separating n u m b e r n o t greater t h a n Q.
A definition of an entirely different kind was proposed by
JOI{DA~,
who drew a t t e n t i o n to the p r o p e r t y possessed b y a limited complete set if the separating number of every pair of existent sets of which it can be expressed as t h e sum is zero. Such a set is, in our sense of the word, united, a n d it is evident fromJORDAN'S work 1 t h a t it is this characteristic t h a t he wishes to emphasise, though he defines it only for closed sets. If we detach the p r o p e r t y which JORDAN de- scribes from the elementary properties which he adds in order to obtain a u n i t e d set, we are led to consider the n a t u r e of a n y set which is such t h a t the separating number of every pair of existent sets of which it is the sum is zero, a n d we have no difficulty in shewing t h a t the sets which have this property are the sets which are connex according to CANTOR'S definition.
23. The Relation between Connex Sets and Congregates.
In a n y universe a set which is a congregate is connex, b u t an unlimited connex set m a y fail to be a congregate, and it is only in a limited universe t h a t every connex set is a congregate.
An example t h a t we have already used will serve again. L e t the universe be a reduced Euclidean plane with distances for separating numbers, let F be a branch of a hyperbola and d an a s y m p t o t e , a n d consider the set obtained by adding F a n d J . If this set is expressed as the sum of two existent sets O, O, either F contains both a point of 0 a n d a point of O, or J contains both a point of 0 and a point of ~, or O coincides with one of the sets F, J a n d q9 coincides with the other. In the first two cases, from the properties of straight lines a n d hyperbolas, it is true both t h a t G ' O a n d G ' ~ have a common point a n d t h a t the separating number of O a n d 9 is zero; in the last case the separat- ing n u m b e r is zero, but the sets are n o t inseparable. Thus there is a division of the whole set into two separable sets b u t there is no division into two sets whose separating number is n o t zero, a n d therefore the set is n o t a congregate b u t b y JORDAN'S criterion it is connex. The same example will serve to illustrate CANTOR'S definition. Of a n y pair of points of the set, either both members belong to F, or both members belong to J , or one member belongs to F, a n d the other to J . If F, z are a n y two points of F or are a n y two points of J , let ~ , de- note the length of the curve or line between y and z, and if # is a n y signless real number let [~] denote the integer n e x t below o (which is o ~ I if o is itself an integer). Then if r is a n y signless n u m b e r other t h a n zero a n d if y a n d z are distinct points of F or distinct points of J , the curve or line between y a n d z can be divided into [~z/r + I parts of equal length, a n d the end points of these parts, [~ym/Q] + 2 in number, compose a set of the required form; if, how- ever, with the same value of r we have to consider two points y, z of which y
Above all, from his phrase d'un seul tenant.
belongs to /~ a n d z to .Y, the first step is to select a point t, of T and a point w of J whose distance a p a r t is not greater t h a n ~, a selection t h a t is possible because d is an a s y m p t o t e of F , a n d we can then find a set of the required kind with the finite number
[;tu,,/Q ] + [;~,,/a]
+ 4 of points.We have said t h a t a limited connex set is necessarily a congregate. Since in practice mathematicians have dealt almost exclusively with limited sets, it is impossible at present to pronounce on the importance of connexity when possessed by a set which is not a congregate. B u t there is no d o u b t t h a t in m a n y cases it is easier to apply the criterion of inseparability t h a n to utilise either CANTOR'S criterion or JORDAN'S, and I think it is safe to p r e d i c t t h a t for logical developments the hypothesis t h a t the separating number of two sets is zero will prove to be conveniently analysed into the form t h a t either the sets are insepar- able or t h e y are separable b u t their separating number is zero a n d one at least of t h e m is unlimited.
I t should be added t h a t the idea of congregates as distinct from connex sets is n o t necessary to the definition of united sets, since in t h a t definition closed congregates only are used a n d there is no extensional difference between closed congregates and closed connex sets. F o r the t h e o r y of sets of points it might seem more elegant, because more economical, to define united sets by means of connex sets a n d n o t by means of congregates, b u t for the course we have adopted there is a double justification t h a t the definition of a congregate lends itself more readily t h a n the definition of a connex set to symbolic treat- m e n t and t h a t while the notion of a connex set is inapplicable except to sets of points the idea of a congregate can be extended to a n y universe in which ag- gregates have derivatives of their own type. Here concludes our digression.
24. Continuous Sets and Continua.
A s t u d y of the formal definitions reveals t h a t every unit set is united.
This result is in fact desirable, b u t there are m a n y properties of multiple united sets which do not belong to unit sets: to mention the simplest, a multiple united set is dense, as is a n y multiple congregate, b u t a u n i t set is not dense. On account of the differences of which this is a typical example, a distinctive epi- t h e t is given to a united set which has more t h a n one unit: a multiple u n i t e d set is said to be continuous.
There appears to be no agreement among mathematicians as to the use of the word continuum; in simple cases, to some writers a c o n t i n u u m is essentially
