366 MATHEMATICS: R. L. MOORE 
pose that it is not. Then there exists an infinite sequence P, Pi, P 2 , P3, . . . 
of distinct points belonging to M and a sequence of positive numbers e, e h 
e 2 , e 3 , . . . such that 1) e/2 > ei > e 2 > e 3 > . . . . 2) Lim. w==0 o e n = 0, 
3) the distance from P n to P is equal to e n , 4) if « is a positive integer there 
exists no connected sub-set of M containing P n and P such that all the points 
of this sub-set are at a distance of less than e from P. Let K denote a circle 
with center at P and radius e/2. The point-set M contains a closed connected 
subset t n that contains P n and at least one point on K but no point without 
K. There does not exist an infinite sequence of distinct positive integers 
m, n 2 , m, . . . such that, for every m, t~ m has a point in common with t\. 
For if such were the case then P + t\ + t„ + tii 2 + • • . would be a con- 
nected point-set containing Pi and P and lying entirely within a circle with 
center at P and radius e. It follows that there exists a positive integer n\ 
greater than 1 such that if n ^ «i then / n has no point in common with t\. 
Similarly there exists n 2 greater than n\ such that if n ^ n 2 then t n has no 
point in common with t n . This process may be continued. It follows that 
M contains an infinite sequence of closed connected point-sets t n ^ tn t , t n , . . . 
{n\<n 2 <n 3 < . . . ) such that no two of them have a point in common and 
such that, for every m, t nfn contains P nm and at least one point on K but no point 
without K. Let K denote the circle whose center is P and whose radius is e\. 
For every m, t n contains a closed, connected subset such that 1) every point 
of tm is either on K or K or between K and K, 2) } m contains at least one point 
on K and at least one point on K. Let K* denote a circle concentric with 
K and K and lying between them. There exists on K* a point O and a sequence of 
points Oi, 0 2 , Ozj . . . such that O m belongs to tm and such that O is a limit 
point of the point-set Oi + 0 2 + 0$ + • ■ • • Let K r denote a circle lying 
between K and K and with center at O. Let 2h denote its radius. Let K" de- 
note a circle with the same center and with a radius less than h and also less 
than 5k/ 2. There exist within K" four points O m{) Om 2 , Om 3 and Om 4 of the set 
O), 0 2 , 0 3 , . . . . Of these four points, two, Ai and A 3 , are separated from 
each other on K* by the other two, A 2 and A±. For each i (1 g i ^ 4) let 
flj denote that one of the point-sets t m , tm o , tm a and t m to which the point Ai be- 
longs. For each i (1 ^ i ^ 4) there exists, within (fig. 2), a circle i^i, with 
center at Ai, that neither encloses nor contains a point of any of the sets ai, 
a 2 , as, a i} except # t . The circles Ki and K 3 respectively enclose points Pi and B s 
belonging to P. Every closed connected point-set that contains both Pi 
and B s and lies between K and K must contain a point either of a^ or of a 4 . 
But there exists a connected subset of P that contains both Pi and P 3 and 
lies wholly with K' . Thus the supposition that M is not connected imkleinen 
has led to a contradiction. 
Since M is limited, closed, connected and connected im kleinen, it follows 7 
that every two points of M can be joined by a simple continuous arc lying 
wholly in M. The point-set M contains a countable subset TV of points 
