34 
MATHEMATICS: R L. MOORE 
Proc. N. a. S. 
ABCD and EFGH denote two squares concentric with I and lying within 
it and such that EFGH is within ABCD, AB and EF being horizontal. 
Let I and / denote points in the face / such that the planes lAB and J CD 
are parallel to b and such that I is equidistant from A and B, while / 
is equidistant from C and D, We thus have a right prism whose triangular 
. bases are ABI and DC J. Let P denote this prism and let Q denote the 
right pyramid whose base is the square EFGH and whose apex is at 0, a 
point half-way between I and /. For each positive integer n, let a„ de- 
note a plane parallel to / and at a distance from / equal to a/n, where a 
is one-half the length of a side of K. For each n, let jS„ denote a plane paral- 
lel to, and half-way between the planes q;„ and a^-i-i- In denote that 
portion of the interior of P which is intercepted between the planes and 
Qn. Let R denote the interior of the pyramid Q. Let Si denote the 
point-set composed of R together with the sets Ii, 1% h, I4, • • • • Let N 
denote the point-set composed of the surface of K together with its interior 
and let M denote N-R. The set M is a continuous curve, Si is one of its 
complementary domains, and the straight line interval // is a portion of 
the boundary of Si. But no point of JI, except O, is accessible from any 
point of Si. The boundary of Si is not connected im kleinen at any point 
of ]I except O. Clearly S2, the exterior of the cube K, is uniformly 
connected im kleinen. 
2. If e is any preassigned positive number, there exists a continuous 
curve M suck that there are infinitely many distinct domains, complementary 
to M and all of diameter greater than e. 
Example. — Consider again the cube K and the domains Ii, I-i, Is,-- - of 
Example 1, it being understood that the distance between the planes TAB 
and JCD of that example is 2e. Let M denote the point-set — (Ii + 72 -f- ^3 
+ . . . . ) where N has the same meaning as in Example 1 . Here M is a con- 
tinuous curve and each I„ is one of its complementary domains. 
3. There exists, in three dimensional space, a continuous curve M which 
divides space into just two domains. Si and S2, such that (a) every point of 
M is a limit point both of Si and of S2, (b) not every point of M is accessible 
from a point of S2. 
Example. — Let K denote a square whose vertices A, B, C, D Site the 
points (0, -1, -1), (0, 1, -1), (0, 1, 1) and (0, -1, 1), respectively. For 
each integer n (positive or negative) let K„ denote a square whose ver- 
tices, An, By,, C„, Dn, are the points {1/n, -1, -1), 1, -1), {l/n, 
1, 1) and {l/n, —1, 1), respectively. For each n let Kn denote a square 
whose vertices, An, Bn, Cn, 'Dn, are so situated that An is half-way be- 
tween ^„ and An,, 5„ is half-way between J5„ and Bni,Cn is half-way be- 
tween C„ and Cn, and Dn is half-way between Dn and Dn,, where ni de- 
notes w + 1 or n—1 according as n is positive or negative. For every 
integer w and every pair of integers i and / which are numerically less 
