MATHEMATICS: R. L. MOORE 
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last member. The point-set M is covered by the family of point-sets com- 
posed of all the members of the sequence ^2 but there does not exist any mem- 
ber g of (32 such that M is covered by the family composed of g together with all 
those members of ^2 that precede g. But this is contrary to Theorem 1. 
Thus the supposition that ^1 is not the first member of /5 has led to a contra- 
diction. It follows that gi is the first member of (3 and that M is covered by 
the finite set of point-sets gj, g2, gs, . . . gn. Thus the Heine-Borel-Le- 
besgue Theorem holds true in every class 5*. 
Suppose now that the Heine-Borel-Lebesgue Theorem holds true in a given 
class S and that G is a monotonic family of closed, compact subsets of S. 
Let g denote any point-set of the family G. I will show that the members of 
G have at least one point in common. Suppose that this is not the case. Then, 
if P is a point of g, there exists a closed point-set of the family G that does 
not contain P. Hence, by Lemma 1, the point P is in the interior of some 
point-set Rp which contains no point of gp. Let H denote the set of all Rp^s 
for all points P of ^. By hypothesis, g is covered by a finite subfamily 
Rp^yRp^jRp^, . . . Rp^ of the family H. But there exists an i (l^i^n) 
such that gPi is a subset of each of the point-sets gp^, gp^, gp^, . . . gp^. 
It follows that no point of gPi is in any one of the sets Rp^, Rp^, Rps, . . . Pp„. 
Thus the supposition that there is no point common to all the members of G 
has led to a contradiction. 
" §2. I now raise the question whether it is not desirable to substitute, for 
Frechet's definition of the word compact, a definition which is, for some 
spaces, (but not for spaces V) more restrictive than that of Frechet. I will 
say that a monotonic family of point-sets is proper if there is no point that 
is common to all of its members. I will say that a set of points M is compact 
in the new sense^ if for every proper monotonic family F of subsets of M there 
exists at least one point which is a limit point of every point-set of F. 
Suppose that in a space L the infinite point-set N is compact in the new sense. 
The set N contains at least one countably infinite sequence of distinct points 
Pi, P2, P3, . . . For each n let tn denote the point-set Pn, Pn+i, Pn+2, . . . 
The family of point-sets /i, /o, ^3, • • . is a proper monotonic family. Hence 
there exists a point P which is a limiting element of every one of these point- 
sets. It follows that if in a space L a point-set is compact in the new sense 
then it is also compact in the sense of Frechet. That the converse is not true 
for every space 5 may be seen with the help of the following example. 
Example. Let a be a well-ordered set of elements such that (1) if K is 
any countable subset of the elements of a then there exists an element of a 
that follows all the elements of K, (2) if P is a given element of a the set of all 
those elements of a that precede P is countable. If a and h are two non-con- 
secutive elements of a such that a precedes h then the set of all those elements 
of a which follow a and precede b will be called a segment. An element P of 
a will be said to be the limit of a countable sequence Pi, P2, P3 . . . of ele- 
