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I regret to announce that there is a serious error in my paper in these Transactions, volume 84, pp. 430-443. The error was discovered by Louis Kokoris who found that on line 8 of page 434 the expression given as 4[g(bz)](az) should have been 4[g(az)](bz). As a consequence the computation of P(z, g, az, b) yields nothing, the proof of formula (30) is not valid, and the important Lemma 9 is not proved. Thus the paper does not give a proof of its major result stated as Theorem 1. Nevertheless, the theorems of the paper are all correct and we shall provide a revision of the proof here. This revised proof has been checked by Louis Kokoris to whom the author wishes to express his great thanks. We observe first that the equation

The reading of Ayres' interesting paper suggested to me the following remarks: 1. The order of a subset of a set S in a point p4 cannot surpass the order of S in p. Hence if S2 denotes the set of all points of S of order 2, then S2 has in each point of S the order 2, the order 1, or the order 0, where the terms "order 0" and "0-dimensional" are used synonymously. SI(M), S2(1), S" may denote the set of all points of S in which S2 has the order 0, 1, 2, respectively. The points of order 2 of S are also called the ordinary points of S, and the set S2 of all ordinarv points of S may be called the ordinary part of S. The set S" of all ordinary points of the ordinary part of S may be designated the ordinary kernel of S. WVe have

The purpose of this note is to present a new proof of a generalized form of Day's fixed point theorem. The proof we give is suggested by the work of T. Mitchell in his paper, Function algebras, means, and fixed points, [2]. The version of Day's theorem which we present here has not appeared explicitly in the literature before, and seems especially well suited for application to questions concerning fixed point properties of topological semigroups. 1. Preliminaries. We adopt the terminology and notation of [2] except where otherwise specified. New terminology will be introduced as needed. Let y be a convex compactum (compact convex set in a real locally convex linear topological space E), and let A( Y) denote the Banach space of all (real) continuous affine functions on Y under the supremum norm. Observe that A(Y) contains every function of the form h=f\Y + r where fe E* and r is real; thus A(Y) separates points of Y.

1. Denote by G the group of the equation upon which depends the determi. nation of the curves of order n 3 having simple contact at 1 n ( n -3 ) points with a given curve C of order n having no double points. The case in which n is odd was discussed in the former paper (Transactions, January, 1902) and G was shown to be a subgroup of the group defined by the invariants 43, 04, , , * *, the latter group being holoedrically isomorphic with the first hypoabelian grouip on 2p indices with coefficients taken modulo 2. For n even, G is contained in the group H defined by the invariants 04' 069 * with even subscripts. JORDAN has shown (Traite, pp. 229-242) that H is holoedrically isomorphic with the abelian linear group A on 2p indices with coefficients taken modulo 2. The object of the present paper is to establish the latter theorem by a short, elementary proof, which makes no use of the abstract substitutions [al, 1 ., p, p1] of JORDAN, and which exhibits explicitly the correspondence t between the substitutions of the isomorphic groups.

The analogies that exist between infinite series and infinite integrals are well known and have frequently served to indicate the extension of a theorem or a method from one of these domains of investigation to the other. According to a principle of generalization that has been formulated by E. H. Moore, the presence of such analogies implies the existence of a general theory which incltudes the central features of both the special theories.t It is the purpose of the present paper to develop the fundamental principles of that sectioll of this general theorv which contains as particular instances the theories of Cesaro and H6lder summability of divergent series and divergent integrals. Furthermore, the usefulness of the theory will be illustrated by proving a general theorem in it which includes as special cases the Knopp-Schnee-Ford theoremt with regard to the equivalence of the Cesaro and Holder means for summing divergent series, an analogous theorem due to Landau ? concerning divergent integrals, and a further new theorem with regard to the equivalence of certain generalized derivatives. The general theorem just mentioned can be extended to the case of multiple limits so as to include other new theorems, analogous to those referred to above, with regard to multiple series, multiple integrals, and partial derivatives. This extension, however, involves formulas that are considerably more complicated than in the case of simple limits. I shall therefore reserve it for a second paper, as I wish to avoid algebraic complexity in this first presentation of the general theory. Following the terminology introduced by E. H. Moore, we indicate the basis of our general theory as follows

In the paper with the above title, which appeared in these T r a n s a c t i on s, vol. 2 (1901), p. 428, I obtained a number of general theorems concerning the zeros of functions of the form 02 y' ol y, where y is a solution of a hornogeneous linear differelntial equation of the second order. There s a further general theorem of the same character which I should have included in that paper if I had discovered it at the time, and which I now give as VII' below, it being a counterpart to VII in the earlier paper. I also give three special applications of this general result, numbered VIII', VIII", VIII"', since they are counterparts of VIII. A part of VIII' was obtained by Fite in the Annals of M athematics for June, 1917, (see the closing lines of his article) by another method; and it was this special result of Fite's that suggested to me the much more general Theorem VII'. What follows should be regarded as an addendum to the former paper, to be inserted on page 434 at the end of ? 2. All references are to that paper, to which the reader should turn for an explanation of the notation and a statement of the restrictions placed on the functions. VII'. If none of the six functions

In an earlier paper t with the same title, we have defined and studied the properties of certain subsets of a continuous curve? which we call the arc-curves of the continuous curve. In a recent paper, G. T. Whyburnil has defined the cyclic elements of a continuous curve, and he has considered a continuous curve as composed of its cyclic elements and has given a large number of the properties of connected collections of cyclic elements. On examining the two papers it is found that arc-curves and connected collections of cyclic elements have many properties in common; and, in fact, in part II of the present paper we shall show that, although these two sets were defined very differently, every connected collection of cyclic elements of a continuous curve is an arc-curve of the continuous curve, and conversely, every arc-curve that contains more than one point is a collection of cyclic elements of the continuous curve. In part III we will develop some new theory concerning the basic sets of a continuous curve, which were defined in Arc-curves, first paper, and shall show the relation between the basic sets and the nodes of a continuous curve. In part IV we shall show that an irreducible basic set of a continuous curve resembles 'in its properties the set of all end points of the continuous curve. All point sets considered in this paper are assumed to lie in a metric, separable, locally compact space. Notation. We shall use the common notation of the theory of sets, such as A +B, A-B, A B, etc., in its usual meaning. If H is a point set, the symbol H denotes the point set consisting of the points of H together

It has been brought to my attention by Mr. N. E. Steenrod that the lemma of page 708 in the paper referred to is in error. The final assertion of the proof is false. It is therefore necessary to point out that the paper is not "disturbed" by this fault. For if one requires that the Pn, «= 1,2, • • •, of the lemma be arcs then the (altered) lemma does hold, since it is true that the connected sum of a perfect continuous curve and an arc is "perfect." One verifies that this restricted lemma is sufficient for the uses of the paper.

Walsh's Theorem II states that if the roots of f (z) are in the circular region C, and if z is exterior to C, then ai lies in C. This is precisely the result obtained by Laguerre as given on page 59 of volume I of his collected works, if expressed in non-homogeneous coordinates as on page 57. We shall use another form of Laguerre's Theorem (loc. cit., p. 57) to prove the underlying proposition on which Walsh bases his proof of his Theorem I, as follows: Every circle through any point z and its "derived point" a as defined by (1) either passes through all the roots off ( z ), or else has at least one root in the region interior to it, and at least one root in the exterior region. I have recently called attentiont to the fact that this is a corollary of a theorem of Bocher's on jacobians which has served as a starting point in Walsh's earlier papers. With a slight change in Walsh's notation, whereby we substitute z's for a's, the proposition from which Theorem I is deduced may be stated thus

It is the object of this paper to obtain theorems concerning the stability of the straight line solutions, and equilateral triangle solutions, respectively, in the problem of three bodies by means of the theorems and calculations of two preceding papers.t It is shown that the generalized theorems of Bohl can be applied to a neighborhood of the straight line solutions, with arbitrary masses, and to a neighborhood of the equilateral triangle solutions, if the masses are such that the characteristic exponents of the generating orbit are not all pure imaginaries. The mutual distances of the three masses are assumed constant on the generating orbit, in both cases.

This paper contains a number of disturbing misprints: Equation (2) p. 560 should read Gpf(x) = aoxPfm+ a1xPf11 + *** + amixpf + amx. Line 17 p. 561 read Ap(x) XBp(x) instead of Ap(x)Bp(x). The term perfect (volkommen) in Theorem 1 is used in the sense of Steinitz, Algebraische Theorie der Korper, edited by Hasse and Baer, pp. 50-51. Line 21 p. 562 should read Fp(x) = Qp(x) X (xP ax) + Ax. Equation (9) p. 562 should read A = aoa(Pm-l)/(P-1) + aia(P'1)/(P-1) + + am,2a0P+1 + am-,a + am. In the expression line 9 p. 563 the last term should be A(")x. Equation (17) p. 564 should read F'"(x) = Fnl(x)P -Fn-1(Wn)PFn-1(X). Line 8 from below p. 574 should read

This gives the desired result. It might be pointed out that Theorem 4 shows that Ilfn-gn I-->O In ?4 it is tacitly assumed that the measurable set E can be partitioned into measurable sets E,,. This is always the case in separable spaces. To proceed without this assumption it will not be necessary to assume that J is metric. The class So(E) is defined as the class of functions finitely valued on E. Such a function is one for which there is a decomposition of E into a finite number of disjoint measurable subsets on each of which it is constant. This basis necessitates only a slight rewording in a few places. In Lemma 1 the set E should be taken as a set in A. In Theorem 2 the words "functions uniformly continuous" should be replaced by "functions finitely valued." In the proof of Theorem 11 the sentence "Fix . . . continuous on e" should be worded "Fix e with 18(E-e)

Let A8 be the domain in the s-plane (s =o+it) defined by -gl(a) -Ah. The proofs of Theorems 1, I1, and III are the same, with the new function S(oI5.

The purpose of this paper is to prove a taming theorem for a codimension one manifold that is locally flat at some point and has 1-ULC complement. We also prove that any two sufficiently close locally flat embeddings of a codimension one manifold are ambient isotopic. Since this paper was first submitted, R. Daverman has shown that, given any point on a codimension one manifold with 1-ULC complement, some neighborhood of that point lies on a codimension one sphere that is locally flat at some points and has 1-ULC complement. Hence the two papers combined prove that a codimension one manifold is locally flat if and only if its complement is 1-ULC.

Let F be either a field of char 0 with all roots of unity or a field of char p > 0. Let G be a countable group. Then all simple NIG]-modules are injective if and only if G is locally finite with no elements of order char F and G has an abelian subgroup of finite index. The condition that all simple modules over a ring be injective first appeared in a theorem due to Kaplansky: a commutative ring satisfies the condition if and only if it is von Neumann regular. Several people have studied the property for noncommutative rings, a recent example being [3]. The authors of that paper suggest the problem of characterizing group algebras with this condition. In this paper we make substantial progress by offering Theorem 3. Let F be either a field of char 0 with all roots of unity or a field of char p > 0. Let G be a countable group. Then all simple FtG]-modules are injective if and only if G is locally finite with no elements of order char F and G has an abelian subgroup of finite index. The proof is divided into three parts. In ? 1, we show that F is injective as an 14G]-module if and only if G is locally finite with no elements of order char F. In the second and crucial section, we show that for a certain class of rings ("locally Wedderburn algebras") the condition that all simple modules are injective is equivalent to the property that all simple modules are finite dimensional over their commuting rings. In -?3, we prove the main theorem by showing that if all simple modules are finite dimensional over their commuting rings then G is abelianby-finite. We would like to thank D.S. Passman for suggestions that shortened and improved our work. 1. Results of Villamayor. Most of this section can be gleaned from several of Villamayor's papers. However, the statements and proofs that appear here are new. The field F is a right 14G]-module under the trivial action (i.e. if k E F, then k * a = k(a)e where E is the augmentation map). We characterize those G with F being 14G]-injective. By considering maps from ideals of FtH] to F, the reader can casily prove Received by the editors February 5, 1973. AMS (MOS) subject classifications (1970). Primary 16A26, 20C05; Secondary 16A52, 16A64.

Let C denote the category of compact Hausdorff spaces and H : C → H C H:C \to HC be the homotopy functor. Let S : C → S C S:C \to SC be the functor of shape in the sense of Holsztyński for the projection functor H. Let X be a continuum and H n ( X ) {H^n}(X) denote n-dimensional Čech cohomology with integer coefficients. Let A x = char H 1 ( X ) {A_x} = {\text {char}}\;{H^1}(X) be the character group of H 1 ( X ) {H^1}(X) considering H 1 ( X ) {H^1}(X) as a discrete group. In this paper it is shown that there is a shape morphism F ∈ Mor S C ( X , A X ) F \in {\text {Mor}_{SC}}(X,{A_X}) such that F ∗ : H 1 ( A X ) → H 1 ( X ) {F^\ast }:{H^1}({A_X}) \to {H^1}(X) is an isomorphism. It follows from the results of a previous paper by the author that there is a continuous mapping f : X → A X f:X \to {A_X} such that S ( f ) = F S(f) = F and thus that f ∗ : H 1 ( A X ) → H 1 ( X ) {f^\ast }:{H^1}({A_X}) \to {H^1}(X) is an isomorphism. This result is applied to show that if X is locally connected, then H 1 ( X ) {H^1}(X) has property L. Examples are given to show that X may be locally connected and H n ( X ) {H^n}(X) not have property L for n > 1 n > 1 . The result is also applied to compact connected topological groups. In the last section of the paper it is shown that if X is compact and movable, then for every integer n, H n ( X ) / Tor H n ( X ) {H^n}(X)/{\operatorname {Tor}}\;{H^n}(X) has property L. This result allows us to construct peano continua which are nonmovable. An example is given to show that H n ( X ) {H^n}(X) itself may not have property L even if X is a finite-dimensional movable continuum.

Two properties of B-convexity are shown to hold for P-convexity: (1) Under certain conditions, the direct sum of two P-convex spaces is P-convex. (2) A Banach space is P-convex if each subspace having a Schauder decomposition into finite dimensional subspaces is P-convex. 0. Introduction. In the previous paper [1] the question of whether all B-convex spaces are reflexive was discussed. The concept of a P-convex space was introduced by C. Kottman [4] as follows: Definition. For a positive integer n, let P(n, X) be the supremum of all numbers r such that there is a set of n disjoint closed balls of radius r inside U(X)= fx: llxll < 11. X is said to be P-convex if P(n, X) < Y2 for some n. Kottman showed that all P-convex spaces are both B-convex and reflexive. Therefore the question "Is there a B-convex space that is not P-convex?" is of interest. Many properties of B-convex spaces are not known for P-convex spaces. In this paper we consider two of these properties and prove partial analogs of them for P-convex spaces: The first property is that direct sums of B-convex spaces are B-convex [2]. The proof of this fact for B-convex spaces rests on the invariance of B-convexity under isomorphism, but it is not known whether P-convexity possesses this invariance. Two partial analogs of the direct sum property are obtained, Theorems 1.3 and 1.5, using Ramsey's theorem of combinatorics. The second property is that a space is B-convex if each subspace having a basis is B-convex [1]. A partial analog of this is proved, Theorem 2.1, using one of the direct sum results. We will use the following characterization of P-convexity from Remark 1.4 of [4]: Let a set of n elements be called 5 separated of order n provided the distance between any two elements of the set is at least &. Then a space X is P-convex if and only if for some positive integer n and some positive number e < 2 there is no 2 e separated set of order n in U(X). Received by the editors July 7, 1971. AMS (MOS) subject classifications (1970). Primary 46B10; Secondary 46B05, 46B15, 05A05.

The countable box product of ordinals is examined in the paper for normality and paracompactness. The continuum hypothesis is used to prove that the box product of countably many a-compact ordinals is paracompact and that the box product of another class of ordinals is normal. A third class trivially has a nonnormal product. Because I have found a countable box product of ordinals useful in the past [1], this class of spaces particularly interests me. The purpose of this paper is to tell what I know about which of these spaces is paracompact or normal. In [2] I prove that the continuum hypothesis implies the box product of countably many a-compact, locally compact, metric spaces is paracompact. I prove here that the continuum hypothesis implies the box product of countably many a-compact ordinals is paracompact (Theorem 1) and the box product of another class of ordinals is normal (Theorem 2). The proof of Theorems 1 and 2 is a quite messy join of the techniques of [1] and [2] which raises some doubt in my mind as to whether these theorems are worth proving. Because I care, because I think these spaces are set theoretically interesting and topologically useful, because I think these theorems are best possible, the theorems are worth the mess to me. A. If {XXAOA)\ is a family of topological spaces, a box in HIAeA XA is a set TheA UA where each UA is open in XA. The box product of {XAIA A iS HAA XA topologized by using the set of all boxes in it as a basis. Throughout the paper the following notation is used. An ordinal a is the set of all ordinals less than a and a is topologized by the interval topology. The statement that a is a cardinal means that a is an ordinal and no smaller ordinal has the same cardinality as a. The notation IAnA BAA is used to mean the ordinary Cartesian product of the f,3's and never the cardinal or ordinal arithmetic product. Similarly a#9 means the set of all functions from ,B into a rather than an arithmetic operation. If a is an ordinal, let cf(a) denote the cofinality of a; that is cf(a) is the smallest ordinal 8 such that there is a subset A of a, order isomorphic with 8, such that /3 < a implies there is a y El A with /3 < y. Observe that a is a a-compact ordinal if and only if a is compact or cf(a) =w. Received by the editors December 10, 1971 and, in revised form, October 1, 1972. AMS (MOS) subject classiflcations (1970). Primary 54B10, 54A25, 54D15, 54D20, 54D30, 02K25.

This paper continues the study of the smoothness properties of (real) topological linear spaces. First, the smoothness results previously obtained about various important classes of locally convex spaces, such as Schwartz spaces, are improved. Then, following the ideas of Bonic and Frampton, we use these results to give sufficient conditions for the existence of smooth partitions of unity on manifolds modelled on topological linear spaces. 1. Preliminaries. In order to make this paper self-contained we include the definition of the two types of derivative used and the definition of an 8-category. We will employ the definitions of the derivative in topological linear spaces investigated in detail by Averbukh and Smolyanov [2], [31. See also [14], [15] and [191. Let TLS denote the class of all Hausdorff topological linear spaces over the real field R. Let 2 1(E, F) = 2(E, F) denote the set of all continuous linear maps from E into F, where E, F e TLS. We define by induction p(E, F) = 2(E, p 1(E, F)). Each ?1p(E, F) is given the topology of uniform convergence on bounded subsets of E. If X is a topological space, W(X) will denote the class of all open subsets of X. Let f: U -' V, where U e ((E), V E C(F), E e TLS and F c TLS. Then we say f is Fre6cbet differentiable at x e U, if there exists u e S(E, F) such that for each bounded subset B of E and for each o-neighbourhood W in F, there exists 8 > 0 such that f(x + th) f(x) u * th E tW, whenever b E B and ItL