210 
MATHEMATICS: R. L. MOORE 
merits of a if, and only if, for every segment ^ that contans P there exists a 
positive integer dps such that, for every n greater than 8ps, Pn is in s. With 
respect to this conception of limit of a sequence, the elements of a evidently 
constitute a class S. The set M composed of all the elements of a is compact 
in the sense of Frechet. It is not, however, compact in the new sense. For if 
for every element x of a, denotes the set of all those elements of a which 
follow X then there exists no element which is a limiting element of every 
member of the proper monotonic family composed of all /^'s for all elements x 
of a. The set M, though closed and compact in the sense of Frechet, does not 
possess the Heine-Borel-Lebesgue property. 
It is easy to see that, in every class V of Frechet, a point-set which is com- 
pact in the Frechet sense is also compact in the new sense. 
By a proof in large part identical with the above proofs of Theorems 1 and 
2, the truth of the following theorem may be established. 
Theorem 3. In a class S, in order that a point-set M should possess the Heine- 
Borel-Lehesgue property it is necessary and sufficient that M should be compact 
in the new sense and closed. 
1 Frechet, M., Palermo, Rend. Circ. Mat., 22, 1906, (1-72). 
2 By a countable sequence is meant a sequence of the same order type as the sequence of 
positive integers arranged in the normal order. 
3 Frechet, M., Bid. sci. math., Paris, 45, 1917, (1-8). See also Chittenden, E. W., Bull. 
Amer. Math. Soc, New York, 25, 1918, (60-65). 
4 In the present paper the elements of a class L will be called points. 
5 A Class F is a class L in which there exists a distance function. Cf. either of the above 
mentioned papers of Frechet. 
^Hedrick, E. R., Trans. Amer. Math. Soc, New York, 12, 1911, (285-294). Cf. also 
Frechet, loc. cit. 
' For a proof that the elements of any set (whether countable or uncountable) can be 
arranged in a well-ordered sequence, see Zermelo, E., Math. Ann., Leipzig, 59, 1904, (514) and 
65, 1908, (107-128). Zermelo assumes the truth of the well-known Zermelo Postulate. With 
regard to this postulate, see a recent paper by Ph. E. B. Jourdain, Paris, C. R. Acad. Sci., 166, 
1918, (520 and 984). 
^ Every subset of a well-ordered sequence contains a first member, that is to say, a mem- 
ber that precedes every other member of that subset. 
^ It is clear that in every space 5 a point-set is compact in the new sense if and only if, 
it possesses the property K defined in §1 of the present paper. 
1" For a proof of the existence of a well-ordered sequence satisfying these two conditions 
cf. Hobson, E. W., The Theory of Functions of a Real Variable, Univ. Press, Cambridge, 1913, 
(177-181). 
Sometime after the manuscript of the prtsent paper had left my hands I found that, 
in 1912, S. Janiszewski introduced an extended conception of limit and defined a point-set 
as ^'parjaitement compact, si de toute de ses elements on pent extraire une suite du meme 
type d'ordre et possedant un eiemente limite." cf. J. ec. polytech., Paris, 16, 1912 (155). 
Compare also the example in §2 of the present paper with an example of Janiszewski's on 
page 167, loc. cit. It seems likely that in a class S a set is compact in my sense if and 
only if it is parfaitement compact in the sense of Janiszewski. I do not find however 
that Janiszewski has made any study of the Heine-Borel-Lebesgue Theorem in connection 
with his conception of compactness. 
