36 
MA THEM A TICS: R. L. MOORE 
Proc. N. a. S. 
:r = 2, J = 0, 7 = 2, s = 0, 2 = 2. For each positive integer n let yl„ denote the 
point (1/w, 1/n, 0) and let denote the point {l/n, 1/n, 2). Let M 
denote the point-set composed of the surface of K together with all the 
straight line intervals Ai Bi, Ai B2, AsB^,. . . . 
5. There exists, in three dimensional space, a closed, hounded and con- 
nected point-set M which is not a continuous curve hut which divides space 
into just two domains Si and S2 such that (a) every point of M is accessihle 
both from every point of Si and from every point of S2, (b) the domain Si is uni- 
formly connected im kleinen. 
Example. — Returning to Example 3, let H denote a cube whose upper 
base lies in the plane z = l, whose center is at the point (0, 0, —1), and 
one of whose faces is in the plane x = 2. For every positive integer n, 
the upper base of the parallelopiped P„ of Example 3 is a part of the upper 
base of H but the remainder of P„ is wholly within H. Let R denote the 
set of points composed of (a) the exterior of H, (h) the interiors of the 
upper bases of the parallelopiped Pi, P2, P3, • • • • , (c) the interiors of Pi, 
P2, P3, . . • • and (d) the interiors of those cylinders Sijn (of Example 3) 
for which n is positive, together with the interiors of the bases of those 
cylinders. Let M denote the boundary of R. The point-set M is closed, 
connected and bounded and it divides space into just two domains. Si 
and S2 (where 5i is R), and furthermore every point of M is accessible 
from every point of 5i and from every point of 52. But M is not a con- 
tinuous curve. It fails to be connected im kleinen at the point (0, 0, 0). 
The domain Si is uniformly connected im kleinen but the domain 52 is not. 
Theorem 1. If, in a three dimensional space 5, the closed, bounded and 
connected point-set M divides 5 into just two domains Si and S2 such that 
(a) every point of M is a limit point both of Si and of S2, and (h) both Si 
and S2 are uniformly connected im kleinen; then M is a continuous curve. 
This theorem will be proved with the assistance of the following lemma. 
A proof of this lemma will be given elsewhere. 
Lemma. Suppose that, in space of three dimensions, (i) M is a closed 
point-set, (2) Ai Bi and A2 B2 are two simple continuous arcs each of which 
consists of a finite number of straight line intervals, (3) neither of the arcs 
Ai Bi and A2 B2 has a point in common with M, {4) every simple continuous 
arc which joins a point of Ai Bi to a point of A2 B2 contains at least one point 
of M. Then M contains a closed and connected point-set which contains at 
least one point in common with each of the straight line intervals Ai A2 and 
Bi B2 and has a diameter less than, or equal to, twice that of the point-set 
composed of the two arcs Ai Bi and Bi B2. 
Proof of Theorem i. — Suppose, on the contrary, that M is not connected 
im kleinen. Then there exists a positive number e and two infinite se- 
quences of points Ai, A2, Az, . . . .and Bi, B2, Bz,...., belonging toM, such 
that Lim An Bn = 0 and such that, for no n, do A„ and B„ lie together 
