F. A. Valentine in [4] proved the following two theorems.

THEOREM 1. Let S be a closed connected subset of Rd which has at most n points of local nonconvexity. Then S is an Ln+i set.

THEOREM 2. Let S be a closed connected subset of Rd whose points of local nonconvexity are decomposable into n closed convex sets. Then S is an L2n+i set.

These results have been extended by a number of authors, but always with stronger hypothesis. (See [1] and [2].) Using a minimal arc technique, new pr∞fs of Theorems 1 and 2 were given in [3].

Valentine remarks in [4] that Theorem 2 might be improved in the case that 5 is the closure of an open connected set. The goal of this paper is to give such an improvement for sets satisfying a particular local connectivity property.

Let be a family of subsets of a topological space X. We do not require to be a covering of X, nor do we assume that the members of are necessarily open. In this paper we shall assume that is of a special sort, which we call Σ-Finite. We show that a Σ-Finite family is both locally finite and star-finite, and in particular that an open covering of X is Σ-Finite if and only if it is star-finite. We then prove that every Σ-Finite family is ᓂ-discrete, so that in particular, every star-finite open covering of X is (ᓂ-discrete. There seems to be some applications of this fact.

A finite line transitive affine plane A is a finite plane which admits a collineation group G acting transitively on the set of all lines of A. Wagner [11] has shown that A is a translation plane and Hering [9] recently investigated the structure of A under the assumption that G has a composition factor isomorphic to a given nonabelian simple group. The purpose of this paper is to show that if the number of points on a line of A is odd, and if G contains no Baer involutions, then the hypothesis of Hering's Main Theorem holds.

It has been widely conjectured that there exists a homotopy commutative Diagram

where J is the stable Whitehead f-homomorphism and BSJ is the space constructed in [3]. In [4], Stasheff and the author proved that this conjecture is false. However, Quillen's proof of the Adams conjecture in [7] has as a corollary the existence of the homotopy commutative diagram

This paper is concerned with the convergence of sequences of subsets of a topological space, as defined by F. Hausdorff [6]. Such a sequence converges if and only if its limit inferior equals its limit superior, where its limit inferior (respectively, superior) is that set each of whose elements satisfies the condition that each of its neighborhoods has nonempty intersection with all but finitely (respectively, with infinitely) many terms of the sequence.

Ruth R. Struik investigated the nilpotent group , where G is a free product of a finite number of cyclic groups, not all of which are of infinite order, and Gm is the mth subgroup of the lower central series of G. Making use of the “collection process” first given by Philip Hall in [8], she determined completely for 1 ≦ n ≦ p + 1, where p is the smallest prime with the property that it divides the order of at least one of the free factors of G. However, she was unable to proceed beyond n = p + 1.

It is a classical result that the only surface of revolution in Euclidean space E3 which is minimal is the catenoid. Of course the surface is conformally flat, but if Mn, n ≧ 4, is a conformally flat hypersurface of Euclidean space En+1, then Mn admits a distinguished direction [2] (“tangent to the meridians“). Thus we seek to characterize conformally flat hypersurfaces of En+1 which are minimal. Specifically we prove the following

THEOREM. Let Mn, n ≧ 4, be a conformally flat, minimal hypersurface immersed in En+1.

All rings are associative with unity. A ring R is prime if xRy ≠ 0 whenever x and y are nonzero. A ring R is (left) primitive if there exists a faithful irreducible left R-module.

If the group ring R[G] is primitive, what can we say about R? First, since every primitive ring is prime, we know that R is prime, by the following

THEOREM 1 (Connell [1, 675]). The group ring R[G] is prime if and only if R is prime and G has no non-trivial finite normal subgroup.

This is a generalization of the results in [3, § 5]. Some of the proofs presented here are actually the original proofs presented in [3]. Although we can find alternate proofs in the case p = 3, since [3] will not be published for a while yet, we feel that it is worthwhile to present the proof in [3] whenever it carries over in the case p = 3. The results in this paper will be used in the investigation of the quadratic pairs for the prime 3.

If (G ; U, V) is a triad with G a group we define

where [g, u] = gug-1u-1 is the commutator. CG(U, V) will be called the (left) center of U in G modulo V or in brief a (left) C-space. If G is a topological group it will be understood that the topology on CG(U, V) is the relative topology of G.

Throughout this paper we use the Bourbaki [1] conventions for rings and modules: all rings are associative but not necessarily commutative and have a 1; all modules are unital.

Our purpose is to extend and simplify some recent results of Maddox [7], Megibben [8], Enochs [3], and the author [5] on absolutely pure modules by introducing several new dimensions, and using the absolutely pure dimension introduced by the author in [6], This completes some work on character modules and dimension in [5] and [6].

An A -module will be called an FFR-module if and only if it has a resolution by finitely generated free A -modules.

A number of unsolved problems of primal algebra theory concern the existence of certain collections of dependent primal algebras. In [3] E. S. O'Keefe showed that any collection of pairwise non-isomorphic primal algebras of type {n} with n > 1 forms a primal cluster. Recently the author has discovered that if τ is any type containing at least two elements, one of which is > 1, then there are at least two non-isomorphic dependent primal algebras of type τ, except possibly in the case = {2, 0}; this result will appear later.

The main theorems concern the relation between the - compact spaces and the -regular spaces, and their analogues in uniform spaces. In either of the categories of Tychonoff spaces or uniform spaces, let be a class of spaces, let be the epi-reflective hull of se (closed subspaces of products of members of ), let be the “onto-reflective” hull of (all subspaces of products of members of ), and let r and o be the associated functors. Let be the class of spaces which admit a perfect map into a member of . Then, is epi-reflective (and in Tych, = but in Unif, the equality fails); call the functor p.

Un anneau commutatif unitaire est dit réflexif si tous ses idéaux sont divisoriels [3; 6]. On dira qu'un anneau commutatif unitaire et intègre est un anneau de Mori, si ses idéaux divisoriels entiers vérifient la condition de chaîne ascendante [8]. Enfin, on appellera provisoirement, M-anneau, un anneau de Mori, tel que tout idéal engendré par deux éléments est divisoriel. Un anneau noethérien réflexif est un M -anneau; dans la première partie de cette note, nous établissons la réciproque. La troisième partie étudie la clôture intégrale de certains anneaux locaux réflexifs non noethériens.

A directed partially ordered abelian group (G, ≦ ) is a tight Riesz group if for a1, a2, b1, b2 ∈ G with ai < bj, i, j = 1,2, there is an x ∈ G with ai < x < bj, i, j = 1, 2. The open interval topology on G is the topology having as a base the set of all open intervals (a, b) = {x ∈ G|a < x < b}. For any x ∈ G, a neighborhood base at x is the set of all open intervals (x — a, x + a) = x + ( — a, a) for a > 0.

The enumeration of the H-equivalence classes of multiplications on a space is a topic of current interest. In this paper we try to study the H-equivalence classes of multiplications on a CW complex X with finitely many non-vanishing homotopy groups, by using the Postnikov decomposition of X and multiplier arguments [1; 4], This paper presents a way to compute the set of H-equivalence classes of multiplications on X from the knowledge of certain quotient sets of H*(B Λ B, ∑) and some homotopy equivalences of B, where B represents the spaces in the Postnikov decomposition of X, and ∑ denotes abelian groups corresponding to the homotopy groups of X.

Let m be any real or complex number, and let Gm be the group generated by the 2 X 2 matrices A = (1, m\ 0, 1) and B = (1, 0; m, 1), where we use the notation (C11, C12; c21, C22) to denote (by rows) the elements of a 2 X 2 matrix C. Thus, Gm is the set of all finite products (or words) of the form

… Ah(3)Bh(2)Ah(1)

where the h(i) are nonzero integers with h﹛\) possibly zero. If a non-trivial word of this form equals the identity / = (1, 0; 0, 1), then Gm is non-free; otherwise, Gm is free.

A distance function d on a set X is a function X × X → [0, ∞ ) satisfying (1) d(x, y) = 0 if and only if x = y, and (2) d(x, y) = d(y, x). Such a function determines a topology T on X by agreeing that U is an open set if it contains an ∈-sphere N(p; ∈)( = {x: d(p, x) < ∈﹜} about each of its points. Equivalently, F is closed if and only if d(x, F) > 0 for each x ∈ X — F. A topological space is symmetrizable via a distance function d if its topology is determined by d as above, and semi-metrizahle via d if x ∈ Ā is equivalent to d(x, A) = 0.

Let F be a variety (equational class) of algebras. For n ≧ 0, Vn is the variety generated by the F-free algebra on n free generators while Vn is the variety of all algebras satisfying each identity of V which has no more than n variables. (Equivalently, Vn is the class of all algebras, , such that every n-generated subalgebra of is in V.) Note that unless nullary operation symbols are specified by the similarity type of V, V0 is the variety of all one element algebras while V° is the variety of all algebras.