Verónica Martínez-de-la-Vega

Uniqueness of hyperspaces for Peano continua

For a metric continuum X and a positive integer n, let Cn(X) be the hyperspace of nonempty closed subsets of X with at most n components. We say that X has unique hyperspace Cn(X) provided that, if Y is a continuum and Cn(X) is homeomorphic to Cn(Y ), then X is homeomorphic to Y . In this paper we study which Peano continua X have a unique hyperspace Cn(X). We find some sufficient and also some necessary conditions for a Peano continuum X to have unique hyperspace Cn(X). Our results generalize all the previously known results on this subject. We also give some significant examples.

An uncountable family of metric compactifications of the ray with remainder pseudo-arc

Abstract In this paper we construct an uncountable family of metric compactifications of the ray with the remainder being the pseudo-arc, answering a question posed by Marwan M. Awartani.

Open Problems on Dendroids

Publisher Summary This chapter discusses basic concepts and some problems on dendroids. A continuum is defined as a compact, connected, metric space. A dendroid is an arc-wise connected and hereditarily unicoherent continuum. A dendrite is defined as a locally connected dendroid. Even though dendroids are one-dimensional and most of them can be geometrically realized, they have many properties and intrinsic characterizations that are still unknown. This chapter presents a survey of some results and open problems on dendroids. The chapter discusses about B. Knaster who saw dendroids as those continua for which for every ɛ > 0 there exists a tree T and an ɛ-retraction r: X → T (an ɛ-retraction is a retraction such that diam(r−1(t)) < ɛ for every t ∈ T). The chapter presents the concepts that are related to mappings on dendrites, maps onto dendroids, contractibility, and hyperspaces. A discussion on property of Kelley, retractions, means, selections, smooth dendroids, planability, and shore sets is also presented in the chapter.

Product topology in the hyperspace of subcontinua

Abstract Let X be a metric continuum. Let C ( X ) be the hyperespace of subcontinua of X . Given two finite subsets P and Q of X , let U(P,Q)={A∈C(X):P⊂A and A ∩ Q =∅} . In this paper we consider C ( X ) with the topology τ P which have the sets U(P,Q) as a basis. In this paper we show that, for a dendroid X , some topological properties of X are very closely related to the topological structure of ( C ( X ), τ P ) .