Nonempty Family Research Articles

Boundedness in a Quasi-Uniform Space

Although a nontopological concept, boundedness seems to be of considerable importance in a topological space. 'There are many topological problems in which it is essential to be able to make this distinction' (between bounded and unbounded sets) [1]. Boundedness and in particular boundedness-preserving' uniform spaces appear to have applications to topological dynamics [4].In spite of this importance, there have been only isolated attempts at developing the concept. Alexander [1] and Hu [7] tried the axiomatic approach. Hu, for example, calls a nonempty family of sets a boundedness if is hereditary and closed under finite union.

Intersections by hyperplanes

LetA be a finite nonempty family of nonempty disjoint closed and bounded sets in a Banach spaceE which is either separable and the conjugate of some Banach spaceX (i.e.E=X*) or, reflexive and locally uniformly convex. IfC denotes the weak*-closed convex hull of ∪ {A:A ∈A} then the set of points inE ∼C through which there is no hyperplane intersecting exactly one member ofA is discrete (or empty).

Generalized Discrete Valuation Rings

Jategaonkar (5) has constructed a class of rings which can be used to provide counterexamples to problems concerning unique factorization in non-commutative domains, the left-right symmetry of the global dimension for a right- Noetherian ring and the transhnite powers of the Jacobson radical of a right- Noetherian ring. These rings have the following property:(W) Every non-empty family of right ideals of the ring R contains exactly one maximal element.In the present paper we wish to consider rings, with unit element, which satisfy property (W). This property means that the right ideals are inverse well-ordered by inclusion, and it is our aim to describe these rings by their order type. Rings of this kind appear as a generalization of discrete valuation rings in R; see (1; 2).In the following, R will always denote a ring with unit element satisfying (W).

Preference Functions on Measure Spaces of Economic Agents

Given a set of economic agents and a set of coalitions, a non-empty family of subsets of the first set closed under the formation of countable unions and complements, an allocation is a countably additive function from the set of coalitions to the closed positive orthant of the commodity space. To describe preferences in this context, one can either introduce a positive, finite real measure defined on the set of coalitions and specify, for each agent, a relation of preference or difference on the closed positive orthant of the commodity space, or specify, for each coalition, a relation of preference or indifference on the set of allocations. This article studies the extent to which these two approaches are equivalent.

Covering the track of an isotopy

We work in the Polyhedral Category as defined in [3], which consists of polyhedra and polymaps. A polyhedron is a space with a maximal nonempty family of p.l. related triangulations and a polymap is p.l. w.r.t. these triangulations. An ambient isotopy of a polyhedron X is a level-preserving homeo, H: XXI->XXI with HIXX1=1, Ht: X->X for tEI is the homeo induced by HIXXt. An isotopy of Y in X is a level-preserving embedding, F: YXI->XXI, Ft: Y-->X for tEI is the embedding induced by FJ YXt. We say that H covers F if HtFo = Ft for all t I, and we say that H covers the track of F if HtFo Y=Ft Y for all tEEI. Given an isotopy F of Y in X, it is not true that there is always an ambient isotopy of X which covers F or even just the track of F. For example, if we restrict attention to manifolds and proper embeddings, then classical knots of SI in S3 are isotopic embeddings which are not ambient isotopic. Ambient isotopy is rather more useful than isotopy, but isotopies are usually easier to construct. The following problem is therefore of interest. Problem A. Given an isotopy F of Y in X, under what conditions can we cover F by an ambient isotopy of X? For some purposes it is enough to cover the track of F, e.g. when considering knots as subspaces rather than embeddings, i.e. working set-wise rather than point-wise; so we have the weaker problem. Problem B. Under what conditions can we cover the track of an isotopy of Y in X? In [1 ] Zeeman and Hudson give a solution to A for proper embeddings of manifolds. Their condition for coverability is local-unknotting of Fl YXJ for any subinterval JCI, which is always true in codimension ? 3, see [1 ] for the precise definition. In this paper we will be concerned only with Problem B and our main result is that any locally collarable isotopy (definition below) can be track-covered. Local collarability is a necessary condition for track-covering and, in the case of proper embeddings of manifolds, is strictly weaker than local unknotting.

A lemma on maximal sets and the theorem of Denjoy-Vitali

By an ∮-related family ∮ we mean a non-empty family ∮ of elements such that to each element F ∈ ∮ is associated a set R(F) of elements of ∮, called the R-class of F, which contains F. An element G ∈ R(F) is said to be R-related to F. By an R-section S of ∮ we mean a set of elements of ∮ such that for any elements F1, F2 of S either F1 ∈ R(F2) or F2 ∈R(F1). If R(F) = {F} for each F ∈ ∮ then the only R-Sections are the sets {F}. The interesting applications of the lemma proved below are to those cases when there exist R-sections which do not contain a finite number of elements.