F.R. Ruiz del Portal

Generating well-behaved utility functions for compromise programming

The purpose of this paper is to seek utility functions satisfying a weak condition which guarantees that the utility optimum always belongs to the compromise set. This set is a special subset of the attainable or feasible set, which is generated through the application of the well-known operational research approach called compromise programming. It is shown that there are large families of utility functions satisfying this condition, thus reinforcing the value of compromise programming as a good surrogate of the traditional utility optimum.

Shape as a Cantor completion process

Let X and Y be metric compacta, Y embedded in the Hilbert cube Q. For two maps f,g:X→Q the authors define F(f,g):=inf{e>0:f is homotopic to g in the e-neighborhood of Y}, and a sequence of maps fk :X→Q, k∈N, is said to be a Cauchy sequence provided for every e>0 there is a k0∈N such that F(fk,fk′)<e whenever k,k′≥k 0. Such sequences coincide with the approximative maps of K. Borsuk [Theory of shape, PWN, Warsaw, 1975] and represent shape morphisms from X to Y. The function F is not a pseudometric, but defining d(α,β):=lim k F(fk,gk), where the shape morphisms α,β∈Sh(X,Y) are represented by Cauchy sequences (fk),(gk), the authors prove that (Sh(X,Y),d) becomes a complete zero-dimensional ultrametric space, homeomorphic to a closed subset of the irrationals. Among other things, the authors prove that if two compacta X and Y are of the same shape, then for every compactum Z, the spaces Sh (X,Z) and Sh (Y,Z) are uniformly homeomorphic. In the last section, the authors show, for example, that for X compact and Y∈FANR , the space Sh (X,Y) is countable, give several characterizations of various kinds of movability, and translate their results to Z-sets in Q and sequences of proper maps between their complements.

Some duality properties of non-saddle sets

Abstract We show in this paper that the class of compacta that can be isolated non-saddle sets of flows in ANRs is precisely the class of compacta with polyhedral shape. We also prove—reinforcing the essential role played by shape theory in this setting—that the Conley index of a regular isolated non-saddle set is determined, in certain cases, by its shape. We finally introduce and study the notion of dual of a non-saddle set. Examples of compacta related by duality are attractor–repeller pairs. We use the complement theorems in shape theory to prove that the shape of the dual set is determined by the shape of the original non-saddle set.

A topology for the sets of shape morphisms

Abstract We introduce a topology on the set of shape morphisms between arbitrary topological spaces X , Y , Sn(X,Y) . These spaces allow us to extend, in a natural way, some classical concepts to the realm of topological spaces. Several applications are given to obtain relations between shape theory and N -compactness and shape-theoretic properties of the spaces of quasicomponents.

On weak shape equivalences

Abstract We prove that weak shape equivalences are monomorphisms in the shape category of uniformly pointed movable continua ShM. We use an example of Draper and Keesling to show that weak shape equivalences need not be monomorphisms in the shape category. We deduce that ShM is not balanced. We give a characterization of weak dominations in the shape category of pointed continua, in the sense of Dydak (1979). We introduce the class of pointed movable triples (X, F, Y), for a shape morphism F : X → Y, and we establish an infinite-dimensional Whitehead theorem in shape theory from which we obtain, as a corollary, that for every pointed movable pair of continua (Y, X) the embedding j : X → Y is a shape equivalence iff it is a weak shape equivalence.

Ultrametrics and infinite dimensional whitehead theorems in shape theory

We apply a Cantor completion process to construct a complete, non-Archimedean metric on the set of shape morphisms between pointed compacta. In the case of shape groups we obtain a canonical norm producing a complete, both left and right invariant ultrametric. On the other hand, we give a new characterization of movability and we use these spaces of shape morphisms and uniformly continuous maps between them, to prove an infinite-dimensional theorem from which we can show, in a short and elementary way, some known Whitehead type theorems in shape theory.

Fixed Point Index in Hyperspaces: A Conley-Type Index for Discrete Semidynamical Systems

Let X be a locally compact metric absolute neighbourhood retract for metric spaces, U subset of X be an open subset and f:U -->X be a continuous map. The aim of the paper is to study the fixed point index of the map that f induces in the hyperspace of X. For any compact isolated invariant set, K subset of U, this fixed point index produces, in a very natural way, a Conley-type (integer valued) index for K. This index is computed and it is shown that it only depends on what is called the attracting part of K. The index is used to obtain a characterization of isolating neighbourhoods of compact invariant sets with non-empty attracting part. This index also provides a characterization of compact isolated minimal sets that are attractors

Finite approximations to & Cech homology

We show in this paper how to represent intrinsically Cech homology of compacta, in terms of inverse limits of discrete approximations. We establish some relations between inverse limits and non-continuous homotopies and, as a consequence, we get a strong form of the classical continuity property of Cech homology.

Shape invariance of N-compactifications

Abstract We use N -compactifications of 0-dimensional spaces to obtain a new shape invariant for the class of all topological spaces. We also point out that the shape and topological classifications are not the same in the realm of Tychonov spaces having a base of clopen sets. Finally we use the new shape invariant to obtain classes of spaces where the conditions “to have the same shape” and “to be homeomorphic” are equivalent.