We give a characterization of monoidal closed structures, “determined by systems of filters” on the category of topological spaces and continuous maps. The method we use to introduce suitable topologies on the product set X × Y of spaces X and Y, and on the set of all continuous maps from X to Y, is essentially that of Wilker.

Let K be a compact Hausdorff space and let E be a Banach space. We denote by C(K, E) the Banach space of all E-valued continuous functions defined on K, endowed with the supremum norm.

Recently, Talagrand [Israel J. Math. 44 (1983), 317–321] constructed a Banach space E having the Dunford-Pettis property such that C([0, 1], E) fails to have the Dunford-Pettis property. So he answered negatively a question which was posed some years ago.

We prove in this paper that for a large class of compacts K (the scattered compacts), C(K, E) has either the Dunford-Pettis property, or the reciprocal Dunford-Pettis property, or the Dieudonné property, or property V if and only if E has the same property.

Also some properties of the operators defined on C(K, E) are studied.

The purpose of this paper is to give an application of homological methods to game theory. The main theorems give conditions of a homological nature for the existence and uniqueness of the Shapley value for games with continuum players.

A set mapping on a set S is a function f mapping S into the powerset of S such that x ∉ f(x) for each x in S. The set map f has order θ if θ is the least cardinal such that |f(x)| < θ for each x in S. A subset H of S is free for f if x ∉ f(y) for all x, y in H. In this paper we use classical results about set mappings of large order to investigate conditions which ensure a large free set for set mappings of unrestricted order.

Let E = {z: |z| < 1} and let H = {w : regular in E, w(0) = 0, |w(z)| < l, z ∈ E}.

Let P(A, B) denote the class of functions in E which can be put in the form (1 + Aw(z))/(1 + Bw(z)), −1 ≤ A < B ≤ 1, w(z) ∈ H. Let S*(A, B) denote the class of functions f(z) of the form such that zf′(z)/f(z) ∈ P(A, B). If f(z) ∈ S*(A, B) and g(z) ∈ S*(C, D) then, in this paper the radius of starlikeness of order β (β ∈ [0, 1]) of the following integral operator

is determined. Conversely, a sharp estimate is obtained for the radius of starlikeness of the class of functions

where g(z) and F(z) belong to the class S*(A, B).

We treat necessary and sufficient conditions for the classical solvability of the Dirichlet problem for the equation of prescribed Gauss curvature in uniformly convex domains in Euclidean n space. Our methods simultaneously embrace more general equations of Monge-Ampère type and we establish conditions which ensure that solutions have globally bounded second derivatives.

Sufficient conditions are obtained for a homogeneous scalar integrodifferential system to have solutions all of which either converge to zero or oscillate or at least one of which can be nonoscillatory asymptotically.

A classification is begun of varieties of algebras with the property that each algebra in the variety has a modular lattice of subalgebras. This turns out to be a very restrictive condition. Such a variety is hamiltonian. If the algebras in it are idempotent, then it is a variety of sets. A variety is subalgebra-modular if and only if it is hamiltonian and satisfies certain conditions on the terms in its three generator free algebras.

The simple twills on n harnesses can be classified according to their maximum float length. The number of n-harness twills with specified maximum float length is determined both by Burnside enumeration and, for n ≤ 20 , by an adaptation of a sieve algorithm for twills.

We construct a connected subspace M of the euclidean plane R2 containing two points A and B such that, for every pair of points {P, Q} of M\{A, B}, the three real numbers d(A, P), d(P, Q) and d(Q, B) are not the same. This solves a question posed by Väisälä.