272 
MATHEMATICS: R. L. MOORE 
Axiom 4.® Every infinite set of points lying in a region has at least one 
limit point. 
Axiom 5. There exists an infinite set of points that has no limit point. 
Axiom 6/ If R is a region and AB is an arc such that AB-A is a sub- 
set of R then {R-\-A)—AB is a connected set of points. 
Axiom V. Every boundary point of a region is a limit point of the exte- 
rior of that region. 
Axiom 8. Every simple closed curve is the boundary of at least one region. 
The System 23. — The system 23 is composed of Axioms 1/ 2/ 3, 4', 5, 
6/ 7/ and 8, where Axioms V , 2', and 4' are as follows: 
Axiom V. If P is a point, there exists an infinite sequence of regions, 
Ri, R2, Rz, . . . such that (1) P is the only point they have in commofi, 
(2) for every n, Rn+i is a proper subset of Rn, (3) if R is a region contain- 
ing P then there exists n such that Rn is a subset of R. 
Axiom 2' . If A and B are two distinct points of a region R then there 
exists, in R, at least one simple continuous arc from A to B. 
Axion 4'. // R is a region, R^ possesses the Heine-Borel property. 
An example of a system satisfying 22 is obtained if in ordinary 
Euclidean space of two dimensions, the term region is applied to every 
bounded connected set of points M, of connected exterior, such that every 
point of M is in the interior of some triangle that lies wholly in M. 
Details (including a third system of axioms, the system Si) will 
appear in Transactions of the American Mathematical Society, probably in 
April, 1916. 
1 Cf., among others, D. Hilbert, The Foundations of Geometry (translation by E. J. Town- 
send, Chicago, 1902) and O. Veblen, A system of axioms for geometry, Trans. Amer. Math. 
Soc, 5, 343 (1904). 
2 Lennes defines a continuous simple arc connecting two points A and B,A^B,3isa, bounded, 
closed, connected set of points [A ] containing A and B such that no proper connected subset 
of [A] contains A and B. Cf. N: J. Lennes, Amer. J. Math., 33, 308 (1911). 
^ A set of points M is said to be separable if it contains a countable subset K such that 
every point of if is a limit point of K. See M. Frechet, Sur quelques points du calcul fonc- 
tionnel, Palermo, Rend. Circ. Mat., 22, 6 (1906). 
^ For definitions of these terms see below. 
^ A space S is said to be descriptive, or potentially descriptive, if it contains a system of 
open curves such that through every two points of 5 there is one and only one curve of this 
system. 
®In view of a result due to F. Hausdorff, it is clear that in the presence of the other 
axioms of S2, Axiom 4 is equivalent to the axiom that if is a region then R' possesses the 
Heine-Borel property. See F. Hausdorff, Grundziige der Mengenlehre, Veit and Comp., 
Leipzig, 1914. 
