MATHEMATICS: R. L. MOORE 
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that closes down on P, Axiom 2' postulates that every two points of a 
region are the extremities of at least one simple continuous arc that 
lies in that region, and Axiom 4' postulates that every region plus its 
boundary possesses the Heine-Borel property. 
An open curve is defined as a closed, connected^ set of points K such 
that if P is any point of K then K-P is the sum of two mutually exclusive 
connected point-sets neither of which contains a limit point^ of the other 
one. It is proved that every open curve is unbounded^ and divides the 
set of all remaining points into two domains. 
Though 2 3 is a sufficient basis for a very considerable part of plane 
analysis situs, nevertheless there exist spaces that satisfy 23 but are 
neither metrical, descriptive^ nor separable. 
It is interesting that no space that satisfies 23 can be potentially de- 
scriptive without being also separable and potentially metrical. In- 
deed if to 23 there be added the axiom that there exists a system of open 
curves such that through every two points there is one and onl> one 
curve of this system, the resulting set of axioms is categorical with respect 
to point and limit point of a point-set. 
Every space that satisfies 22 satisfies also 23, but not conversely. 
In every space satisfying 22 there exists infinitely many open curves 
through any two given points. I have not however determined whether 
every such space is descriptive. 
Definitions. — ^A point P is said to be a limit point of a point-set M if, 
and only if, every region that contains P contains at least one point of 
M distinct from P. The boundary of a point-set M is the set of all 
points [X] such that every region that contains X contains at least one 
point of M and at least one point that does not belong to M. If If is a 
set of points, denotes the point-set composed of M plus its boundary. 
A set of points K is said to be bounded if there exists a finite set of re- 
gions Ri, R2, Rs . . . Rn such that is a sub-set of (Ri+R^+Rz-^- 
. . . -\-Rn)'. If is a region the point-set S-R^ is called the exterior of 
R. A set of points is said to be connected if however it be divided into 
two mutually exclusive sub-sets, one of them contains a limit point of 
the other one. 
The System 22. — Axiom 1. There exists an infinite sequence of regions, 
Ki, K2, Kz, . . . such that (1) if m is an integer and P is a point, there 
exists an integer n greater than m, such that Kn contains P, (2) if P 
and P are distinct points of a region R, then there exists an integer 8 
such that if n>8 and Kn contains P then Kn is a subset of R-P. 
Axiom 2. Every region is a connected set of points. 
Axiom 3. If R is a region, S-R' is a connected set of points. 
