PROCEEDINGS 
OF THE 
NATIONAL ACADEMY OF SCIENCES 
Volume 8 MARCH 15. 1922 Number 3 
ON THE RELATION OF A CONTINUOUS CURVE TO ITS COM- 
PLEMENTARY DOMAINS IN SPACE OF THREE 
DIMENSIONS'- 
By Robert L. Moork 
Department of Pure Mathematics, University of Texas 
Communicated by E. H. Moore, February 1, 1922 
Schoenflies has shown^ that, in order that a closed, bounded and con- 
nected point-set, lying in a plane S, should be a continuous curve, it is nec- 
essary and sufficient that (1) if is a domain complementary to M, every 
point of the boundary of R should be "accessible from all sides" with re- 
spect to R, and (2) if e is any preassigned positive number, there do not 
exist infinitely many distinct domains, complementary^ to M, and all of 
diameter^ greater than e. In the present paper I will exhibit examples 
of continuous curves, in space of three dimensions, which satisfy neither of 
these conditions. 
If, in a plane S, a closed and bounded point-set K separates S into just 
two domains. Si and 52, such that every point of is a limit point both of Si 
and of 52, then, in order that K should be a simple closed curve, it is^ nec- 
essary and sufficient that every point of K should be accessible® from every 
point which does not belong to K. In this paper it will be shown that, 
for a space 5 of three dimensions, this condition is neither necessary nor 
sufficient in order that K should be a simple closed surface or in order that 
it should be a continuous curve. It will be shown, however, that a closed, 
bounded and connected point-set in space of three dimensions is a con- 
tinuous curve provided it is the common boundary of two mutually ex- 
clusive domains both of which are uniformly connected im kleinen."^ 
1. There exists, in a three dimensional space 5, a continuous curve M 
which divides 5 into just two domains, Si and S2, such that (a) the boundary 
of Si contains a point which is not accessible from any point of Si, (b) the 
domain S2 is uniformly connected im kleinen, (c) the boundary of Si is not 
a continuous curve. 
Example. — Let K denote a cube placed so that b, one of its bases, is 
horizontal. Let / and / denote two opposite lateral faces of K. Let 
