Voh. 8, 1922 
MA THEM A TICS: R. L. MOORE 
37 
in a closed and connected subset of M of diameter less than or equal to e. 
But, since the domains Si and 52 are uniformly connected im kleinen, 
there exists a positive number be such that if X and Y are two points both 
belonging to Si, or both belonging to 52, and such that XY < he then X 
and Y can be joined by a simple broken line of diameter less than e/9 which 
lies wholly in 5i or wholly in 52. There exists an integer w such that A^, 
< Se/S. For each positive integer n there exists spheres and /3„, 
with centers at and B^, respectively, and with diameters less than 
each of the numbers 5e/3, e/9 and 1/n. There exist, within points 
En and E' n belonging to 5i and 52, respectively, and within points 
Dn and D'^ belonging to 5i and 52, respectively. There exist broken lines 
Dn Eyi and D'^ E'n lying in 5i and 52, respectively, and with diameters 
less than e/9. Every simple continuous arc from a point 6f En to a 
point of D'n E'n contains a point of M. It follows from the above lemma 
that M contains a closed and connected point-set M„, of diameter less than 
or equal to 2e/3, w^hich contains a point within a„ and a point within 
The sequence of closed and connected point-sets Mi, M2, M-s, .... has, as its 
limiting set,^ a closed and connected subset of M which contains both and 
Bjn and has a diameter less than e. Thus the supposition that M is 
not connected im kleinen has led to a contradiction. 
1 Presented, in part, to The American Mathematical Society, October 29, 1921. 
2 Schoenflies, A., Die Entwickelung der Lehre von den Punktmannigfaltigkeiten, Zweiter 
Teil, Leipsig, 1908, p. 237. 
2 A connected set of points M lying in a space 5 of one or more dimensions is said to 
be a domain (with respect to S) provided there exists, in S, a closed (or vacuous) point- 
set N such that M = S—N. The domain R is said to be complementary to the closed 
point-set M \i (1) R contains no point of M, and (2) the boundary of is a subset of M. 
^ The diameter of a bounded point-set M is the smallest number d such that if X and 
Y are any two points of M then the distance from X to F is less than d. 
/ Cf. Schoenflies, A., "Ueber einen grundlegenden Satz der Analysis Situs," Gc/Wwgew, 
Nadir. Ges. Wiss., 1902 (185). 
^ A point X, belonging to a closed point-set K, is said to be accessible from a point 
Y which does not belong to K if there exists a simple continuous arc from X to Y which 
has no point except X in common with K. 
^ A point-set ilf is said to be connected "im kleinen" if, for every point P of Mand every 
positive number e, there exists a positive number 5ep such that if X is a point of M at a 
distance less than 5ep from the point P, then X and P lie together in a connected subset 
of M every point of which is at a distance of less than e from the point P. The set M 
is said to be uniformly connected im kleinen if for every positive number e there ex- 
ists a positive number 5e such that if Pi and Pi are two points of iVf at a distance apart 
less than Be then they lie together in a connected subset of M every point of which is 
at a distance of less than e from Pi. Cf. Hahn, H., Wien. Ber., 123, 1914 (2433). Hahn 
shows that in order that a closed, bounded and connected point-set M should be a con- 
tinuous curve it is necessary and sufficient that M should be connected im kleinen. See 
also, however, Mazurkiewiez, S., Fundamenta Mathematicae, 1, 1920 (166-209). In this 
article the author refers to earlier articles by himself, published, in 1913 and 1916, in 
a journal iC. R. Soc. Sc. Varsovie) to which I do not, at present, have access. 
