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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 P and the sequence in question are themselves determined, (2) if the element P is the limit of the sequence P1, P2, P3, . . . and ni, n2, n3, . . is an infinite sequence of positive integers such that n1<n2<N3 . . . then P is also the limit of the sequence Pn,, Png, 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 limiting elements of a given set M is called the derived set of M. A class S is a class L in which the derived set of every set is closed. An element P belonging 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 M of L 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 paper3 Fréchet has shown that in order that, in a given class S, a point-set M should have the Heine-Borel property it is necessary and sufficient 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 M in a class V 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. Fréchet 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 pointsets 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 S in which every compact subset has the property K will be called a class S*.

Lemma 1. If in a class L every set of points which contains the point P in its interior contains at least one point of the set M distinct from P, then the point P is a limiting element of the set M.

Proof. If P were not a limiting element of M then it would be interior to L-M+P. But this set of elements contains no point of M distinct from

P.

Theorem 1. If, in a class S*, M is a closed and compact set of points and B is a well-ordered sequence of point-sets such that M is covered by the family composed of all the members of B, then there exists a member g of the sequence ẞ such that M is covered by the family composed of g together with all those members of B that precede g.

6

Proof. Suppose there exists no such member g. Then for each member x of the sequence ẞ let M, denote the set of all points P belonging to M such that P is not in the interior of x or of any member of ẞ that precedes x. For every x, M, contains at least one point. For every x, M, is closed. For suppose P is a limiting element of Mr. If P were not in M, then it would necessarily be in the interior either of the point-set x or of some point-set of the sequence ẞ that precedes x and therefore, by a lemma of Hedrick's, this particular pointset would contain, in its interior, at least one point of M, which is contrary to the definition of M. It follows that the family composed of all M,'s for all members x of the sequence ẞ is a monotonic family of closed point-sets. Hence there exists a point Po which is in every M. But Po is in the interior of some point-set xo of the sequence B. Hence Po is not in Mo. Thus the supposition that Theorem 1 is not true leads to a contradiction.

Theorem 2. In order that the Heine-Borel-Lebesgue Theorem should hold true in a given class S it is necessary and sufficient that that class S should be also a class S*.

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Proof. Suppose that in a given class S* the closed and compact point-set M is covered by the infinite family G of point-sets. The members of the family G can be arranged in a well-ordered sequence B. By Theorem 1, there exists at least one member g of ẞ such that M is covered by the family composed of g and all those members of ẞ that precede g, together with not more than a finite number of those elements of ẞ that follow g. Let gi denote the first such g and let g2, 83, 84, .. gn denote a finite set of members following g1 such that M is covered by g1, 82, 83, .. gn together with all those members of ẞ that precede gi. There clearly cannot exist a member of ẞ immediately preceding g1. I will show that gi is the first member of 8. Suppose it is not. Then if g1 and all the succeeding members of the sequence 8 are removed, there remains a well-ordered sequence ẞ whose members are the remaining members of 8 arranged in exactly the same order as in the original sequence B. Let us now construct a third well-ordered sequence 2 having for its first ʼn members the point-sets g1, 82, 83, . . . gn arranged in the order indicated and having as its remaining members the members of ẞ arranged in the same order as in B1. The point-set ẞ2 has no

last member. The point-set M is covered by the family of point-sets composed of all the members of the sequence ß2 but there does not exist any member g of B2 such that M is covered by the family composed of g together with all those members of B2 that precede g. But this is contrary to Theorem 1. Thus the supposition that g1 is not the first member of ẞ has led to a contradiction. It follows that g1 is the first member of ẞ and that M is covered by the finite set of point-sets g1, g2, g3, . gn. Thus the Heine-Borel-Le

besgue Theorem holds true in every class S*.

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 ğ 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 gp 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 g. By hypothesis, g is covered by a finite subfamily RP, RP, RP3, RP of the family H. But there exists an i (1 ≤i≤n) such that gp; is a subset of each of the point-sets 8P1, P2, 8P3, ・・・ It follows that no point of gp; is in any one of the sets RP1, RP2, RP3, ... RP. Thus the supposition that there is no point common to all the members of G has led to a contradiction.

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§2. I now raise the question whether it is not desirable to substitute, for Fréchet's definition of the word compact, a definition which is, for some spaces, (but not for spaces V) more restrictive than that of Fréchet. 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 P1, P2, P3, . . For each n let t, denote the point-set Pn, Pn+1, Pn+2, The family of point-sets t1, t2, tз, . . . is a proper monotonic family. Hence there exists a point P which is a limiting element of every one of these pointsets. 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 Fréchet. That the converse is not true for every space S may be seen with the help of the following example.

Example. Let a be a well-ordered set of elements such that 10 (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 b are two non-consecutive elements of a such that a precedes b 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 P1, P2, P3 . . . of ele

ments of a if, and only if, for every segment s that contans P there exists a positive integer dps such that, for every n greater than ôps, P, 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 Fréchet. It is not, however, compact in the new sense. For if

for every element x of a, t, 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 ty's for all elements x of a. The set M, though closed and compact in the sense of Fréchet, does not possess the Heine-Borel-Lebesgue property.

It is easy to see that, in every class V of Fréchet, a point-set which is compact in the Fréchet sense is also compact in the new sense.11

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 HeineBorel-Lebesgue property it is necessary and sufficient that M should be compact in the new sense and closed.

1 Fréchet, 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.

Fréchet, M., Bul. sci. math., Paris, 45, 1917, (1-8). See also Chittenden, E. W., Bull. Amer. Math. Soc., New York, 25, 1918, (60-65).

In the present paper the elements of a class L will be called points.

5 A Class V is a class L in which there exists a distance function. Cf. either of the above mentioned papers of Fréchet.

Hedrick, E. R., Trans. Amer. Math. Soc., New York, 12, 1911, (285–294). Cf. also Fréchet, loc. cit.

7 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).

8 Every subset of a well-ordered sequence contains a first member, that is to say, a member that precedes every other member of that subset.

It is clear that in every space S 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.

10 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).

11 Sometime after the manuscript of the present paper had left my hands I found that, in 1912, S. Janiszewski introduced an extended conception of limit and defined a point-set as "parfaitement compact, si de toute de ses éléments on peut extraire une suite du même type d'ordre et possédant un élémente limite." cf. J. éc. 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.

AN ELECTRODYNAMOMETER USING THE VIBRATION

TELESCOPE

BY CARL BARUS

DEPARTMENT OF PHYSICS, BROWN UNIVERSITY

Communicated, April 21, 1919

1. Introductory.-The employment of a telescope with a vibrating objective did good service as an aid to the interferometry of vibrating systems. It seemed worth while therefore to see what could be got out of it, when used in connection with a telephone only, as a dynamometer. The experiments are of interest both because of the vibratory phenomena observable and in view of the peculiar method of optic observation developed. Its possible use for finding the magnetic field within a helix of unknown constants deserves mention. 2. Apparatus.-A front view (elevation) of the design is given in figure 1 and an enlarged detail (side view) in figure 2. The apparatus consists of a rigid rectangular frame-work of inch gas pipe, A, B, C, D, EE', F being the foot attached to a tripod. There may be a steadying foot at C'. A and D are attached to EE' by the stout clamps c, c', so that EE' lying behind the plane of ABCD, admits of the attachment of a suitable clamp c', by which the telephone ih may be held in the same plane. B and C may be forced apart slightly by the screw n controlled by the broad thumb nut m, the conical end of n rotating in a socket of the cap p.

The vibrating system consists of the bifilar wires of phosphor bronze dd' and the frame of the lens ƒ which is the movable objective of the telescope T, the latter part containing the ocular and a plate micrometer (cm. divided in 100 parts). T may be at a considerable distance (50 cm. or more) from ƒ, and supported by a convenient standard. The frame of the lens (which must hold it securely, cement being used if necessary) is of light sheet metal, the parts gg' being of sheet iron (about 0.05 cm. thick) so as to be attracted by the magnet, i, of the telephone. The stiff cross wires, r,s, of the frame are either soldered to the bifilar system dd' or otherwise attached to it (soft sealing wax does very well for temporary experimental purposes).

The attachment and tension-control of the bifilar suspension is finally to be described, as its period must be synchronized with the alternating current. Results are obtainable only when the two periods are strictly in unison. In figure 1 the wires dd' are looped around a groove in the pipe D below, and the upper ends of dd' after passing a similar groove in A are bent around the post a, a', and wound respectively around the snugly fitting screws, b, b', the ends being secured against sliding by a fine hack saw cut in the screws. To stretch a wire it is passed from the notch in b once or twice around it, thence around a, downward by the groove to D and then up in the corresponding way to b'.

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