Vol. 8, 1922 
MA THEM A TICS: R. L. MOORE 
35 
than 4'"', let Py„ denote that point, within the square whose y and 
z are and //4'"^, respectively. For each let r„ denote the set of all 
points [Pijn\ for which i and / are even numbers numerically less than 
4'^' and let Un denote the set of all points [Pijn] such that i and / are odd 
numbers numerically less than 4'"'. For every point Py„ belonging to 
Tn or to Un let Kijn denote a circle lying in the plane Kn> with center at 
T'ijn and with a radius equal to 1/4'"'"^^. Thus with each point of the set 
Tn + Un there is associated a definite circle having that point as center, 
and every one of these circles lies wholly without eveiy other one. Now 
let Pijn and Kijn denote the orthogonal projections of Pijn and Kijn, 
respectively, onto the plane of the square Kn^ and let Py„ denote the orthog- 
onal projection of Pijn onto the plane of 7v„. For each point Pijn be- 
longing to the set T„ let Sy„ denote the right circular cylinder whose 
bases have the circles Kijn and Kjjn as their perimeters, and for each 
point Pijn belonging to Un let Cy„ denote the right circular cone whose 
base has Kijn as its perimeter and whose apex is the point Pijn- It is 
to be noted that Sijn exists only when i and / are even and that Cijn ex- 
ists only when i and / are odd. For each n, let P„ denote the right paral- 
lelopiped which has Kn and Kn as opposite lateral faces and let denote 
the interior of P^. Let denote that portion of which remains 
after the removal of every point of which lies either within, or 
on the surface of any Cy„ {i and / odd). For each n let L„ denote the 
point-set obtained by adding to 7„ the interiors of all the cylinders S^jn 
{i and / even) together with the interiors of the bases of these cylinders. 
Let K* denote a rectangle lying in the plane z = 2 and with sides equal and 
parallel to the respective sides of the upper base of Pi. Let P' denote a 
parallelopiped whose upper base is K"^ and whose lower base is the upper 
base of Pi. Let P'' denote a parallelopiped whose upper base is X* and 
whose lower base is the upper base of P-i. Let L* denote the point-set 
composed of the interiors of P' and P'' and the interiors of the upper bases 
of Pi and P-i. Let H denote the point-set L* -f Li -f L2 + L3 + . . . . , and 
let M denote its boundary. The point-set H is a domain and M is a con- 
tinuous curve which separates space into two domains 5i and 52 (where 
5i is i^) . Every point within the rectangle K belongs to M. But no point 
within this rectangle is accessible from any point in 52, though every point 
of M is a limit point both of 5i and of 52. Neither 5i nor 52 is uniformly 
connected im kleinen. 
4. There exists, in a space 5 of three dimensions, a closed, connected and 
hounded point-set M which divides 5 into just two domains such that (a) 
both oj these domains are uniformly connected im kleinen, and every point of 
M is accessible from one of these domains, but (b) M is not a continuous 
curve. 
Example. — Let K denote the cube which is bounded by the planes x = 0,. 
