CONCERNING THE SUBSETS OF A PLANE CONTINUOUS CURVE A THESIS IN MATHEMATICS PRESENTED TO THE FACULTY OF THE GRADUATE SCHOOL OF THE UNIVERSITY OF PENNSYLVANIA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY HARRY MERRILL GEHMAN Reprint from ANNALS OF MATHEMATICS, Vol. 27, No. 1 PHILADELPHIA 1925 Lsi ^ LOTOKK & WoLFr, RAMBUKO, GKHMANT CONCERNING THE SUBSETS OF A PLANE CONTINUOUS CURVE.* By Habky Merrill German. I. Introduction. In the present paper we shall discuss plane continuous curvest with reference to certain problems concerning the subsets of a continuous curve. It will be understood that the point sets considered lie in a -Euclidean space of two dimensions. In part II of this paper is given, a method for the construction of a continuous curve from any bounded, closed, and connected point-set by the addition of a countable infinity of arcs, thus showing that any bounded, closed and connected point-set may be a subset of a continuous curve. In part III are given two conditions which are necessary and sufficient, and one condition which is sufficient but not necessary, that every closed and connected subset of a continuous curve be a continuous curve. In part IV is given a characterization of simple continuous curves by means of one of the conditions introduced in part in. In conclusion, we wish to express our thanks to Professor John Robert Kline, who first suggested the problems which are discussed in this paper, and who, by his never failing encouragement, has materially assisted in theii' solution. n. In this part we shall consider this problem: given a bounded, closed and connected point-set K, which is not a continuous curve; can we always construct a bounded, connected point-set L, such that K-\-L is a con¬ tinuous curve? If so, what conditions must the set L satisfy? The first question is answered in the affirmative by Theorem IE, while Theorem I and its corollary state that L must be a set which cannot be expressed as the sum of a finite number of arcs. Theorem 1, If K is a hounded, closed and connected point-set, which is not a continuous curve, and if L is any arc, then K-]r L is not a con¬ tinuous curve. Proof. Since K is not a continuous cuiwe, it contains a continu of condensation W, such that K is not connected im kleinen at any point of TT.I * Various parts of this paper were presented to the American Mathematical Society, April 19, October 25, and December 30, 1924. fFor definitions and theorems concerning continuous curves, see E. L. Moore: Report on continuous curves from the vieurpoint of analysis situs, Bull. Amer. Math. Soc., vol. 29 (1928), pp. 289-302. This paper will hereafter he referred to as "Report". t Report: pp. 296-97. 29 30 h. m. gehman. Suppose L is an arc which does not contain every point of W, and let P be a point of W which is not a point of L. Since L is closed, about P we can put a circle C, not containing or enclosing any points of L. Since the only points oi K-\-L which C contains or encloses are points of K only, and since K is by hypothesis not connected im kleinen at P, it follows that -ST-j-P is not connected im kleinen at P, and therefore that K-]-L is not a continuous curve. Suppose L is an arc which contains every point of W. Since W is closed and connected, W is an arc, AB, which is a sub-arc of L. Let P be a point of AB.* Since L—AB is closed, we can put about P a circle C, not containing any points of P — AB. C will contain no points of P that are not also points of K, and therefore, as in the preceding paragraph, jBT-f P is not connected im kleinen at P, and is not a continuous curve. We have shown that no matter how the arc P be constructed, K-\-L is not a continuous curve. The following corollary can be proved by a repeated application of Theorem I. Corollary. If K is a hounded, closed and connected point-set, which is not a continuous curve, and if L is any point-set which can he expressed as the sum of a finite number of arcs, then K-\- L is not a continuous curve. Note that the next theorem shows the necessity of assuming in this Corollary that the number of arcs into which P is separated, is finite., Theorem II. If K is a hounded, closed and connected point-set, which is not a continuous airve, there exists a hounded connected point-set L consisting of a countahle infinity of arcs, such that K-\-L is a continuous curve. Proof. Let us construct in the plane of P' a pair of coordinate axes, and let us select our unit of length so that the diameter of K is greater than 2. Since K is bounded, there is some positive integer n, such that every point of P" is at a distance less than n from the origin. Let Pi be the sum of the straight line intervals from (—n, i) to (n, i), {i — 0, ±1, ±2, • • •, Azn), and the straight line intervals from (J, n) to 0> — 0 = i 2, • • •, it n). Let W denote the set of all points of K at which K is not connected im kleinen, plus all limit points of such points. In each square of unit side formed by intervals of Pi, that contains or encloses a point of TF, let us join the mid points of opposite sides by segments of lines parallel to the X and Y axes. Let Pg be the sum of the segments thus added. In general, let Li {i = 3, 4, 5, • • •), be the set of segments added to P1 + P2+ • • • +Pi-i, by joining mid points of opposite sides of each square ♦If XYZ is an arc, XYZ = XYZ—X — Z. SUBSETS OF A CONTINUOUS CURVE. 31 of side formed by intervals of Ly]-Lz-\- Li-i, that contains or encloses a point of W. Let L = Li + -^2 + • • • +-^1-1- • • • • Since each Li consists of a finite number of arcs, L will consist of a countable infinity of arcs. The set L is bounded and connected. L is not necessarily closed, but every limit point of L which is not in L is a point of W. Therefore L is bounded, closed and connected. The set L (and therefore K-\-L) is connected im kleinen at all points, which are not points of K. The set K (and therefore + L) is connected im kleinen at all points, save points of W, It remains to be shown that is connected im kleinen at every point of W. Let P be a point of W. Let us put about P as center a circle C, of arbitrary radius e. There exists an integer such that 1/2-''2, such that C encloses a square of side 1/2^^ or less, formed from lines of P14-P2+ ••• +P;, which encloses P. The square encloses no points of TF, and therefore the supposition that it encloses a point P of P, contradicts our definition of the construction of P. We have thus proved that, if P is constructed as in Theorem II, it 4* 32 H. M. GEHMAN. follows that, if K is nowhere dense, K-{-L is nowhere dense, while, if K is dense in any parts, K-\-L is dense only in those parts in which K is dense. "We can include Theorem I, its Corollary, Theorem II, and this additional property of Z'+L in a new theorem: Theorem HI. If K is any hounded, closed and connected point-set which is not a continuous curve, a point-set L consisting of a countable set of arcs, necessarily infinite in number, can be so constructed that K-\-L is a continuous curve tvkich is dense only in those parts in tvhich K is dense. m. It is well known that a continuous curve may have the property that each of its closed and connected subsets is also a continuous curve, while another continuous curve may contain closed and connected subsets which are not continuous curves. In the first class is any continuous curve which contains no simple closed curve;* in the second class is any continuous curve which is dense in any part. In this section we obtain necessary and sufficient conditions that a continuous curve be in the first class, i. e., that every closed and connected subset be a continuous curve. We shall first introduce and discuss several definitions. Definition. If M and W are point-sets, then M is said to be connected im hleinen relative IT at a point P, if for every positive number e, there exists a positive number d, such that, if X and Y are any two points of Jlf at a distance less than d from P, then they lie in a connected subset of M, which is such that either (1) every point is at a distance less than e from P, or (2) every point is at a distance less than e from a point of W. Definition. If M and W are point-sets, then M is said to be uniformly connected im Tdeinen relative to W, if for every positive number s,. there exists a positive number d, such that if X and Y are any two points of If at a distance apart less than d, then X and Y lie in a connected subset of M which is such that either (1) it lies within a circle of radius e, or (2) every point is at a distance less than e from a point of TF.t If if is a closed and bounded set and is connected im kleinen relative to W, then it is uniformly connected im kleinen relative to W, but this is not necessarily true in case if is open or unbounded. The property of being relatively uniformly connected im kleinen is weaker * S. Mazurkiewicz, Un theoreme sur les lignes de Jordan, Fundamenta Mathematicae, vol. 2 (1921), p. 123. fThis notion is somewhat akin to that of relatively uniform convergence of functions, due to E. H. Moore. See The New Haven Mathematical Colloquium, SUBSETS OF A CONTINUOUS CURVE. 33 than that of being uniformly connected im Jcleinen, and the property of being relatively connected im kleinen is weaker than that of being connected im kleinen. The properties of being relatively uniformly connected im kleinen and of being connected im kleinen are independent of each other; a set may have either property without having the other. For example, if W is a point on a simple closed curve J and M is J— W, then M is con¬ nected im kleinen but is not uniformly connected im kleinen relative to W. On the other hand, if M is the curve «/ = sin —, for 0 < a? ^ 1 plus the QO interval of the F-axis between y = 1 and y — — 1, and if TF is the portion of the F-axis included in M, then M is not connected im kleinen but it is uniformly connected im kleinen relative to W. For the special case where W is closed and has no points in common with M, the property of being relatively uniformly connected im kleinen is stronger than that of being connected im kleinen. We have given above an example in which M is connected im kleinen but not uniformly connected im kleinen relative to W. If, however, a set M is uniformly connected im kleinen relative to W, where W is closed and has no points in common with M, then M is connected im kleinen. For if it were not, it would fail to be connected im kleinen at a point P. Let e be less than one-third the distance from P to any point of W. For any number d, there are points in if at a distance from P less than d that cannot be joined to P either by a connected set in if which lies within a cmcle of radius e or by a connected set which is such that every point is at a distance less than € from a point of W. The set if is therefore not connected im kleinen relative to W, either uniformly or non-uniformly, contrary to hypothesis. Therefore in case W is closed and has no points in common with if, the property of being relatively uniformly connected im kleinen lies between the properties of being connected im kleinen and of being uniformly connected im kleinen. The property 8 defined by E. L. Moore* is also a property which lies between these two properties. The following examples show, however, that, if W is closed and has no points in common with if, the property of being relatively uniformly connected im kleinen and the property 8 are independent of each other. For if TF is a point on a simple closed curve J, and if is J—TF, then if has property 8, but is not uniformly connected im kleinen relative to TF. If if is the curve y = sin—, for X 0 0, there exists a number di>0, such that if two points of iV—W are at a distance apart less than d,, they can be joined by a connected subset of N—W, which either lies entirely within a circle of radius fi, or is such that every point is at a distance less than from a point of W. They can therefore be joined by an arc in N—W, satisfying one of the two conditions.* If we put about each point of W as center a circle of radius d^/d, by the Heine-Borel Theorem we can select from this infinite set a finite subset of circles, (7i, Cg, • • •, Cn, covering W. Let P,- (^ = 1, 2, • • •, n) be a point ot N—W lying within Q. Since W is connected, each circle of + <^2 + • • • + Cn will intersect at least one other circle. We will aiTange the set of circles in some order, Cn,, Cn,, • • •, Cn^, such that (1) each circle occurs at least once, and (2) each circle intersects both the preceding and the following circles in this arrangement. If Q and Cj intersect = 1,2, • • •, n), Pi and Pj are at a distance apart less than di, and can therefore be joined by an arc in N—W, every point of which is at a distance less than either (2fi-J-4^i) or and therefore less than 3ei, from a point of W, If we construct the arcs, Pn, Pw,, Pw, Pn,, • • •, Pnj_, Pn^, their sum is a continuous curve Ki, having no points in common with W, and such that every point of Ki is at a distance less than 3ei from some point of W, and every point of IF is at a distance less than 3^/2 (and therefore less than fi/2) from some point ot Ki. If 3«2 is less than the distance from any point of W to any point of Ki, we will construct a continuous curve corresponding to in the same way that Ki corresponds to By this process we obtain an infinite sequence of continuous curves, Ki, Ki, Kz, •. •, each one having no points in common with any other one or with W, and such that every point of Ki (^■ = 1, 2, 3, • • •) is at a distance less than 3«i from some point of W, and every point of W is at a distance ' less than fi/2 from some point of Ki. For i^2, < ft_i/6. From these considerations, it follows that the only limit points of (jSTi + -^2 + + • • •) are in W, and every point of TF is a limit point of {Ki + Z'g + -STa + • • •)• * Report, p. 294. See also: R. L. Moore, Concerning continuous curves in the plane, Math. Zeit., vol. 15 (1922), p. 255. SUBSETS OF A CONTINUOUS CURVE. 37 Let P be a point of W. Let a be a positive number such that there are points of W at a distance greater than « from P. Since N is connected im kleinen at P, given a > 0, there exists a number J3>0 such that any point of iV at a distance less than /3 from P can be joined to P by an arc in N of diameter less than a. Only a finite number of the sets, Ki, Ks, Ks, " have no points closer to P than jS, and therefore there exists an integer m such that, for i > m, Ki has points at a distance less than from P. Let Qi {i ^ m) be a point of Ki at a distance less than /J from P. Let us join each of the points, Qm, Qw+i, Qm+2, • • •, to P by an arc in N lying within a circle of radius a. having P as center, which circle we shall denote by C. We will now show that K=W-\- {Km-\-Km+i H ) -|- {Qm P+ Qm4-iP+ • • •) + limit points is a subset of N which is closed and connected, but which is not a continuous curve because it is not connected im kleinen. The set K is connected, by construction, because it consists of a countable infinity of connected sets {W, and (P/ +each of which contains P, forming a connected set which remains connected upon the addition of its limit points. But K is also closed, since we have included in K the limit points of (Q»iP+ • • •)> all of which lie within or on the circle C. Let J. be a point of W exterior to P. If f is less than the distance from A to any point of C, then although there are points of K [i. e., points of {Km-{-Km+i-\- • • •)] arbitrarily close to A, these points can be joined to A only by a connected set in K which lies partly within C, and there¬ fore is of diameter greater than s. Therefore K is not connected im kleinen at A, and is therefore not a continuous curve. Continuing our proof of Theorem IV, we shall next prove that the given condition is sufficient by showing that, if some closed and connected subset K of a continuous curve M is not a continuous curve, then M contains an P-set. By means of K. L. Moore's characterization of closed and connected sets which are not continuous curves,* it is evident that K must satisfy the following conditions: there exists a square A BCD in the plane, and a countable infinity of closed and connected sets, W, Ki, K^, K^, • • •, such that (1) each of these sets is a subset of K, has at least one point on AD and at least one point on BC, has no points on .4P or CD, and is a subset of the set H consisting of ABCD and its interior, (2) no two of these sets have a point in common, and no one of them (save possibly W) is a proper subset of any connected point-set which is common to K and Ht * Report, pp. 296-97. 38 H. M. GEHMAN. (3) the set W is the sequential limiting- set* of the sequence of sets, Ki, Ki, Kz, •' •. From (1) and (2), if K' is any one of the sets, W, Ki, Kz, Kz, ■ then H—K' is disconnected into at least two connected parts, of which one contains AB and a second contains CD, and no connected part save these two contains any point of W Kx-{- Kz-\- Kz-\ . In the case of H—TF, either Hx, the connected part containing AB, or Hz, the connected part containing CD, contains an infinite number of sets of Kx, Kz, Kz, • • •. Suppose Hx contains an infinite number. Let be a point of W interior to ABCD. If we put a circle of radius e about E as center, e being chosen so that the circle is interior to ABCD, then, since if is a continuous curve, there exists a number d, such that any point of M within a circle of radius <5 about E as center can be joined to E by an arc in M within the circle of radius e. By (3) the circle of radius 8 contains a point jP of Ka, where Ka is one of the sets, Kx, Kz, Kz, • • •, which lie in ffi. The arc EE in M and within the circle of radius e contains as a subset an arc Ex Fx such that Ex is in W, Fx is in Ka, Ex Fx has no points in common with either W or Ka, Ex Fx lies, except for Ex, in Hx. Let Nx be the maximal connected subset of M lying in Hx and con¬ taining Fx. Evidently Nx will contain Ex Fx, the set Ka, and an infinite number of other sets of Kx, Kz, Kz, • • •. Since any limit point of Hx not contained in Hx is bontained in W, all limit points of Nx which are not in Nx are in W. Since Nx contains an infinite number of the sets: Kx, Kz, Kz' • •, and since W is their sequential limit, it follows that every point of TF is a limit point of Nx. Therefore Nx + IF forms a closed set, which we shall designate by N. We shall now show that N is an jR-set. N is closed and connected, and TF is a continu of condensation of N. We shall next prove that N—TF is uniformly connected im kleinen relative to TF. Let us select a positive number e. From (3), corresponding to e there is a positive integer n such that, for i'> n, every point of Ki and every point of that connected part 8 of Hx — Ki which has limit points in TF * The point-set Jf is said to be the limiting set of the sequence of point-sets • • • provided that (a) each point of M is the sequential limit point of an infinite subsequence of some sequence of points, Pi, Ps, Ps, • • •, such that, for every n, P» belongs to Mn, (b) if Pi, Pa, Ps, • • • is a sequence of points such that, for every n, P« belongs to Mn, then M contains the sequential limit point of every subsequence of Pi, Pa, Ps, • • • that has a sequential limit point. If the further condition is satisfied that every infinite sub¬ sequence of the sequence, Mx, Ma, Ms, • • •, has the same limiting set M, then M is said to be the sequential limiting set of the sequence Mi, Ma, Ms, • • •. subsets of a continuous curve. 39 is at a distance less than e from some point of W. Let a number i be chosen such that i > n and such that Ki is in iV. We shall denote S^Ki by H^. The set N—W is connected, because N—W — Nx, and Nx is connected by definition. Therefore any two points X and Y oi N— W m can be joined by a connected set in N—W. If this connected set has points in Hx — Hz, the removal of Ki disconnects it into the subsets lying in Hz—Ki and in Hx—Hz. By Lemma A, the subset in Hz—Ki and the set Ki form a connected set joining X and Y in Hz. Therefore any two points in Hz can be joined by a connected set in Hz, and therefore such that every point is at a distance less than s from a point of W. In the set 8 = Hz—Ki let us construct an arc CiA such that Cj is on BC, Di is on AZ>, and CxDx is interior to ABCD. The arc CVA and the arcs DxA, AB, and BCx of ABCD form a simple closed curve J which contains or encloses all points of Ki, and which contains W in its exterior. Let L be the maximal connected set of M lying in J plus its interior and containing Ki. Since J plus its interior is a subset of Hx, L is a subset of N. By Lemma B, L is a continuous curve. Since L is a continuous curve, L is uniformly connected im kleinen. Therefore corresponding to our given f, there exists a positive number such that, if the distance between any two points of L is less than dj, the two points lie in a connected subset of L which lies in a circle of radius «. No point of L save points on CxDx can be a limit point of points of N-^L. Therefore there is a positive number dg which is less than the distance from any point of Ki to any point of N—L. Note also that all points of N—L lie in iS'+ir= Hz — Ki-\-Wand none lie in Hx—Hz. We shall now show that, if d is selected so as to satisfy the inequalities dt (i = 1, 2, • • •) being more than e. An arc CxA can be passed through all the points of Cx,Cs,---, and an arc DxB can be passed through Di, Dg, • • •.* Moreover, these arcs can be drawn so that they have no points in common and so that they have only Cx and Dx in common with the arc CxDx of M. Each of the arcs CsDs, QDs,---, of if contains a subarc, BsFs, E^Fs,---, such that Ei is on CxA, Fi is on BxB, and EiFi has no other points in common with CxA or DxB. From the set of arcs Es Fs, E^Fs, • • •, we can select a subset En, Fn„ En^ Fn^, • •. such that, if Ej Fj and Ek Fn are any two distinct arcs of the subset, then either EjFj is interior (save possibly for its end points) to the simple closed curve Ek Fk Dx Cx, or Ek Fk is interior (save possibly for its end points) to the simple closed curve Ej Fj Dx Cx. Let E he a, limit point of the set En,, En„, - • - . The point E will be on the arc CxA. Let Ox, O2, • • - be a subset of the set En„ En^, • • • which * R. L. Moore and J. R. Kline, On the most general plane closed point-set through which it is possible to pass a simple continuous arc, Ann. of Math., vol. 20 (1919), p. 218. SUBSETS OF A CONTINUOUS CURVE. 41 approaches U as a, sequential limit point, and such that Gi lies between Oj and E on the arc Ci A, if ^' < i. Except possibly for a finite number of points, the corresponding points • • • of the set F^, F^, F^^, • • • will occur on D^B in the same order as the points Oi, O^, • • ■ occur on C^A, Suppose this holds true for all points after Hn. If jP is a limit point of the set Hn, Hn^x, • • •, it cannot lie between any two of them, nor can Hn lie between F and -Hn+i, and F is therefore a sequential limit point of Hn, Jy»+i, • • • and Hi lies between Hj and F on the arc D^B if 7i Jc, an arc GiF in M lies within the circle. Let N be Gk Hk + Gk+x Hk+x + • • • plus GkF-\- Gk+x P-f • • • plus W plus limit points otGkF-\- Gu+x F-\ all of which lie within the circle about F. The set W is a closed and connected subset of M, which is therefore a continuous curve. But N is not connected im kleinen at F because, if a circle is put about F having no points in common with the one about F, it will contain points of N arbitrarily close to F, these points being points of Gk Hk + Gk+x Hk+x + • • •, and, since each of these arcs is distinct and has no points in common with W, points on different arcs can be joined * L. Zoretti, Sur les fonctions analytiques uniformes, Journ. de Math., 6* ser., vol. 1 (1905), p. 10. 42 H. M. GEHMAN. in N only by connected sets having points within the circle about E and therefore exterior to the circle about F. AVe have thus obtained the desired contradiction and the proof of Theorem V is completed. Theorem VI. If M is a continuous curve, a sufficient {hut not necessary) condition that every closed and connected subset of M he a continuous curve is that, given any positive number e, M can he expressed as the sum of a finite number of closed and connected point-sets each of diameter less than e, and each pair of sets having at most a finite number of points in common. Proof. Let M — Mx-\-Mi-\- \-Mn, each a closed and connected set of diameter less than e, and let the points of M common to two or more of these sets be Pj, Pg, • • •, Pfc. We shall determine the maximum possible number of mutually exclusive closed and connected sets of diameter greater than € that are contained in M. Any such set does not lie entirely in any one of the sets Mi, M^, ■ ",Mn and, being connected, must contain at least one of the points Pi, Pg, • • •, P/c. No two of the sets of diameter greater than e can contain the same point. Pi, as the sets are mutually exclusive. Therefore h is the maximum number of such sets. Therefore, by Theorem V, every closed and connected subset is a con¬ tinuous curve. The following example shows that the condition is not necessary. Let Bq be the straight line intervals from (0,0) to (2,0), from (2,0) to (1,1), and from (1,1) to (0,0). Let P, be the straight line intervals from (1,0) to (i) ^); from (1, 0) to (H, i). Let Pg be the straight line intervals from (i, 0) to (i, I); from (J, 0) to (|, |); from (1^, 0) to (IJ, i); from (1^, 0) to (If, i). In general, the intervals of Pi (i = 1, 2, 3, • • •) form with the intervals of P0 + P1 + hPi-i a set of 2^' isosceles triangles, each of which is such that (1) the base is on the X-axis and is of length l/2^~^, (2) the ic-codrdinates of the end points of the base occur among the numbers 0 1 2 3 2* and —1, (4) the ^/-codrdinate of the vertex is 1/2^. Given Bt, Pi+i is the set of 2^"+^ intervals formed by joining, in each triangle, the mid point of the base to the mid point of each side. Let Ai = Pi -f Pg -f P3 -f- • •.. Let Ag be the set {x', y') obtained by subjecting the points (x, y) of Ai to the transformation x' = 2,2/' = —y. Let Af = Ai + Ag. (See Figure.) That every closed and connected subset of M is a continuous curve is best seen by means of Theorem IV. We shall next show that there exists SUBSETS OP A CONTINUOUS CURVE. 43 no finite set of points in M such that M can be expressed as the sum of a finite number of closed and connected sets each of diameter less than 1/2 and each pair having in common only points of this finite set. Suppose such a finite set of points did exists in M. Call them Pi, P^, Pk' On the straight line interval from (^1^2,0) to (2,0), which is common to Ai and A, there occur at least two points of the set, say Pi and P^. If the ic-coOrdinate of Pi is not of the form a/2^, where a and h are integers. Pi is the sequential limit of a sequence of isosceles triangles in Ai, whose bases contain Pi, each triangle in the sequence containing •and enclosing the triangles that follow. Similarly, unless the a:-co5rdinate of Pi is of the form al2^-\-\V^2, where a and h are integers. Pi is the sequential limit of a similar sequence of triangles in A^. Since the O'f'e.-i) ic-coOrdinate of Pi cannot be of both these forms, one of the above cases holds for any point Pi. Suppose in this case that the ar-coOrdinate of Pi is not of the form a!2^. One of the sets contains Pi and an interval, of the Z-axis having one end at Pi and the other end between (0, 0) and Pi. Call this set Mx. One of the sets, say AU, contains Pi and an interval of the .S^axis having one end at Pi and the other end between Pi and (2 + ^1^2,0). Since by hypothesis the rr-cobrdinate of Pi is not of the form a/2^, there is an infinite sequence of triangles in Ax, one end of whose base lies in Mx, and the other end in ilfg. If a is a number less than the distance from Pi to any of the points Pg, P3, • • •, P/c, there is a triangle in Ax of diameter less than « whose base contains Pi and whose other two sides join a point of ilfi to a point of by an arc not containing any points of Pi + PaH + P&. Therefore Mx = Mi. 44 H. M. GEHMAN. If the ic-coordinate of Pi is also not of the form a/2^+il^2, Pi is interior to a simple closed curve in M of diameter less than a, and therefore entirely in Mx. In this case none of the sets M^, M^, • • •. Mn contain Pi. _ If the ir-coordinate of Pi is of the form a/2^+|V'2, Pi is an end point of two straight line intervals of slope -fl and—1 in As. Denote one of these by Pi Q. Let Ms denote the set which contains Pi and an interval of PxQ having one end at Pi. There is an infinite set of line intervals in As, perpendicular to Pi Q, joining Pi Q to the X-axis, and approaching Pi as a sequential limit. As before, there are an infinite number of these at a distance less than a from Pi and therefore joining a point of ilfi to a point of Ms by an arc not containing any points of Pi -f Pg + ■ • • + P/c. Therefore Mi = Ms. Therefore, if Mi contains Pi, it contains an interval of each of the straight line intervals in As having an end at Pi. In this case also there is a minimum distance from Pi to a point of any of the sets Ms, Ms, • • •, Mn, and therefore none of them contain Pi. Therefore Pi can be eliminated from our finite set of points and Pg + Ps + • • • will serve as well. Similarlj'- we can eliminate any other' point lying on the interval from V2, 0) to (2, 0), in which case this interval lies entirely in one of the sets, say Mi. Mi is then of diameter greater than 1/2, contrary to hypothesis. This shows that the condition is not necessary. IV. In part III we have introduced the idea of a point-set being uniformly connected im Jcleinen relative to another point-set. This new idea gives us a characterization of the class of point-sets which E. L. Moore has called simple continuous curves,* namely the simple continuous arc, the simple closed curve, the ray, and the open curve,—the point-sets which are in continuous (1 — 1) correspondencet respectively with an interval of a straight line, a circle, a half-line, and a line. Theorem VII: If M is a closed and connected plane point-set, such that M—W is uniformly comiected im Jcleinen relative to W, where W is any closed and connected subset of M consisting of more than a single point, then M is a simple continuous arc, a simple closed mrve, a ray, or an open curve. Proof. We have shown in part HI that if M—W is uniformly connected im kleinen relative to W, where W is closed, then If—W is connected im kleinen. Since W is arbitrary, it follows that M is connected im kleinen * R. L. Moore, Concerning simple continuous curves, Trans. Amer. Math. Soc., vol. 21 (1920), pp. 333-47. t A correspondence T which sends M into TiM) is said to be continuous if, in case the point P of JIf is a limit point of N, a subset of JUT, then P(P) is a limit point of T{N). subsets of a continuous curve. 45 at every point, and is therefore a continuous curve. There are two cases to be considered. Case I. M contains a simple closed curve J. Since by hypothesis M—J is uniformly connected im kleinen relative to J, no point of J" is a common limit point of a set of points of M interior to J and a set exterior to J. If M contains an arc PQR, such that P and P are on J, and all other points of the arc are either interior to J or exterior to J, then M contains three arcs, PQR, PSR, PTR, having* only P and R in common. One arc, say PSR, lies within the domain bounded by the other two. Since M—{PQR-}-PTR) is uniformly connected im kleinen relative to {PQR-\-PTR), and since P is a limit point of points on the arc PSR interior to the simple closed curve {PQR-}-PTR), there exists a positive number ^i, which is less than the distance from P to any point of M exterior to the simple closed curve {PQR-}-PTR). In the same way we can show the existence of positive numbers and which are less respectively than the distance from P to any point of M interior to {PQR-}-PSR) and the distance from P to any point of M interior to {PSR-}-PTR). Let us now select a positive number a which is less than the smallest of the numbers ti, €2, €2 and also less than the distance from P to R. Let us also select a point U, on the arc PSR, such that the diameter of the arc PZ7 of PSR is less than a/4. We shall now show that M—PTJ is not uni¬ formly connected im kleinen relative to PU. For, if we let e — uf\, no matter what value d is given we can always select a point X, different from P on PQR, and a point Y, different from P on PTR, such that the distances from P to X and P to F are less than d/2 and also less than a/4. The distance from X to Y is less than d, and yet any connected subset N of M containing X and Y either contains R or contains a point not on PQR, PSR, or PTR (in either case, N contains a point whose distance from P is more than a) and therefore N cannot lie within a circle of radius a/4, nor is every point of N at a distance less than a/4 from a point of PU. We have thus shown that the assumption that the arc PQR exists, leads to a contradiction. If there are points of M not on J, and there are no arcs in M joining two points of J, save arcs that lie wholly in J, then any point Q of M—J lies in a maximal connected subset P of M — J, which is closed save for a single limit point P on J. Let R be another point of J, and let us select a positive number a which is less than the distance fi'om P to R. Let us also select a point P of P such that an arc PU exists in P of diameter less than a/4. Then we shall show that M—PU is not uni¬ formly connected im kleinen relative to PP by a proof exactly the same as in the preceding paragraph, save that N in this case necessarily con- 46 h. m. gehman. tains B. We have thus shown that the assumption that there are points of M not on J leads to a contradiction. Therefore, in case I, M = J. Case II. M contains no simple closed curve.* Let AB be an arc in M. It A B contains a point P, A^ P ^ B, which is a limit point of M—AB, and if T is any maximal connected subset of M—AB having P as a limit point, then M—(P-fP) is disconnected, one maximal connected subset of M—(P+P) containing AP — P, another containing PB—P. Since P is a common limit point of these two subsets of M—(P+P), it is evident that M—(P+P) is not uniformly connected im kleinen relative to (P+P); for if we select X and Y according to our previous method, there is no connected subset of M—(P+P) containing both X and Y. We have thus shown that the assumption that the point P exists leads to a contradiction. The points A and B may be limit points of M—AB so that M consists of a set of arcs having only their end points in common. Therefore, in case 11, If is a simple continuous arc, a ray, or an open curve, and the theorem is proved. In conclusion, I wish to point out that the theorem is not true if M—W is uniformly connected im kleinen relative to W is replaced by either the weaker condition, M—W is connected imkleinm, or the independent con¬ dition M—W has the S-property. For example, under these hypotheses M might consist of three arcs having only their end points in common. *For a discussion of this type of continuous curve see S. Mazurkiewicz, loc. cit., pp. 119-130, and K. L. Wilder, Bull. Amer. Math. Soc., vol. 29 (1923), p. 118. Univeesity op Pennsylvania, Philadelphia, Pa.