MATHEMATICS: R. L. MOORE 
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has a definite meaning and the question whether this statement is true or 
false has a determinate answer as soon as the element F and the sequence in 
question are themselves determined, (2) if the element P is the limit of the 
sequence Pi, P2, P3, . . . and ni, n2, ns, . . . is an infinite sequence of 
positive integers such that ni<n2<nz . . . , then P is also the limit of the 
sequence P«,, Pn^, Pn^, ■ • • , (3) if P is an element of L, P is the limit of the 
sequence P, P, P, . . . , whose elements all coincide with the element P. 
An element P is said to be a limiting element of a sub-class M of the class L 
if P is the limit of some infinite sequence of distinct elements belonging to 
M. The set M is said to be compact if every infinite set of distinct elements 
belonging to M has at least one limiting element. The totality of all the lim- 
iting elements of a given set M is called the derived set of M. A class 6" is a 
class L in which the derived set of every set is closed. An element P belong- 
ing to L is said to be interior to the sub-class M of the class L if M contains P 
and at least one element of every sequence of distinct elements that converges 
to P. A family G of sub-classes of a class L is said to cover a subclass M of 
the class L if every element of M is interior to some member of the family G. 
A set of elements M is said to possess the Heine-Borel property if for every 
countably infinite family G of sub-classes of L that covers M there is a finite 
sub-family of G that also covers M. A sub-set if of X is said to possess the 
Heine-Borel-Lebesgue property if for every family G of sub-classes of L that 
covers M there is a finite sub-family of G that covers M. 
In a recent paper^ Frechet has shown that in order that, in a given class S, 
a point-set^ M s^oilld have the Heine-Borel property it is necessary and suf- 
ficient that the set M should be closed and compact. He also shows that the 
same conditions are necessary and sufficient in order that a point set If in a 
class should have the Heine-Borel-Lebesgue property. The Heine-Borel 
Theorem or the Heine-Borel-Lebesgue Theorem is said to hold true in a given 
space L if in that particular space every closed and compact point-set has 
the Heine-Borel property or the Heine-Borel-Lebesgue property respectively. 
Frechet points out that in order that the Heine-Borel-Lebesgue Theorem should 
hold true in a given class L it is necessary, but not sufficient, that the class L 
should be a class S; and sufficient, but not necessary, that it should be a class 
V. He raises the question as to what property it is necessary and sufficient 
that a class L should possess in order that the Heine-Borel-Lebesgue Theorem 
should hold true in that class. In the present paper I will exhibit one such 
property. 
I will call a family G of point-sets a monotonic family if, for every two point- 
sets of the family G, one of them is a subset of the other one. A sub-class M 
of a class L will be said to have the property K in case it is true that for every 
monotonic family G of closed sub-classes of M there is at least one point which 
is common to all the members of the family G. A class 5" in which every com- 
pact subset has the property K will be called a class 5*. 
