MATHEMATICS: R. L. MOORE 
369 
Similarly if Pi, P 2 , P3, . . . is a set of points such that, for every n, P n 
precedes A n on some arc of the set AiBi, A 2 B 2 , . . . , then the set Pi, P2, 
P3, . . . can not have more than one limit point. It is clear that at least 
one of the sets Ai, A 2 , A3, . . . , Pi, B 2 , P 3 , . . . is infinite. Otherwise 
N*, and therefore M, would be a simple continuous arc, which is impossible 
in view of the fact that it is the boundary of a limited domain. There remain 
two conceivable cases. Either 1) the sets A lf A 2 , As, . . . and Bi, B 2 , B%, 
. . . are both infinite or 2) one is finite and the other infinite. Suppose 
they are both infinite. Consider the facts that 1) no set with the properties 
indicated above for P], P 2 , P 3 , . . . has more than one limit point, 2) no 
simple continuous arc is the boundary of a limited domain and, 3)M does not 
contain two arcs with only one common point, that point being an end-point 
of only one of them. In view of these facts it is clear that Ai, A 2 , A3, . . . 
and Bi, B 2 , B3, . . . have a common limit point 0 and that the point-set 
N* + O is a simple closed curve, identical with M . 
Suppose the set Ai, A 2 , A3, . . . is infinite while Pi, B 2 , P 3 , . . . is not. 
With the aid of the same three considerations, it is clear that in this case 
Ai, A 2 , A 3 , . . . has as its only limit point the point B m where m is a positive 
integer such that, for every n greater than m, B n coincides with B m . In this case 
the pount-set TV* is a simple closed curve, identical with M. Of course a simi- 
lar argument applies in case A h A 2 , A3, . . . is finite and Pi, B 2 , P 3 , . . . 
is infinite. Thus in every case M is a simple closed curve. 
I will now proceed to establish the converse proposition that every Jordan 
region is uniformly connected im kleinen. Suppose that, on the contrary, there 
exists a closed curve / whose interior I does not have this property. Then 
there must evidently exist a positive number a, & point 0 on / and two 
infinite sequences of points X h X 2 , X 3 , . . . Y 2 , Y 2 , F 3 , . . . lying in /, 
such that 0 is the sequential limit point of each of these sequences and such 
that for no n can X n be joined to Y n by a connected subset of / that lies en- 
tirely within a circle K with center at O and radius a. But there exists 8 a 
closed curve /* containing O such that every point of /* belongs either to 
/ or to K and such that every point within J* is within both J and K. It is 
clear that for some n the points X n and Y n are both within /*. But the 
interior of /* is a connected subset of I and also of the interior of K. Thus 
the supposition that / is not uniformly connected im kleinen has led to a 
contradiction. 
1 Schoenflies, A., Gottingen, Nachr. Ges. Wiss., 1902, (185). 
2 Kline, J. R., Bull. Amer. Math. Soc, New York, 24, 1918, (471). One of Kline's con- 
ditions is that the common boundary of the domains in question should be connected £ im 
kleinen.' 
3 Caratheodory, C, Math. Ann., Leipzig, 73, 1912-13, (366). 
4 In space of two dimensions, a domain is said to be simply connected if it is connected 
and contains every limited point-set whose boundary lies in it. 
5 A point-set M is said to be connected l im kleinen' if for every point P of M and every 
positive number e there exists a positive number 8 e p such that if X is a point of M at a dis- 
