Displaying 41 – 60 of 240

Showing per page

Coincidence of Vietoris and Wijsman Topologies: A New Proof

Holá, L’. (1997)

Serdica Mathematical Journal

Let (X, d) be a metric space and CL(X) the family of all nonempty closed subsets of X. We provide a new proof of the fact that the coincidence of the Vietoris and Wijsman topologies induced by the metric d forces X to be a compact space. In the literature only a more involved and indirect proof using the proximal topology is known. Here we do not need this intermediate step. Moreover we prove that (X, d) is boundedly compact if and only if the bounded Vietoris and Wijsman topologies on CL(X) coincide....

Consonance and Cantor set-selectors

Valentin Gutev (2013)

Open Mathematics

It is shown that every metrizable consonant space is a Cantor set-selector. Some applications are derived from this fact, also the relationship is discussed in the framework of hyperspaces and Prohorov spaces.

Continua with unique symmetric product

José G. Anaya, Enrique Castañeda-Alvarado, Alejandro Illanes (2013)

Commentationes Mathematicae Universitatis Carolinae

Let X be a metric continuum. Let F n ( X ) denote the hyperspace of nonempty subsets of X with at most n elements. We say that the continuum X has unique hyperspace F n ( X ) provided that the following implication holds: if Y is a continuum and F n ( X ) is homeomorphic to F n ( Y ) , then X is homeomorphic to Y . In this paper we prove the following results: (1) if X is an indecomposable continuum such that each nondegenerate proper subcontinuum of X is an arc, then X has unique hyperspace F 2 ( X ) , and (2) let X be an arcwise connected...

Continuous selections on spaces of continuous functions

Angel Tamariz-Mascarúa (2006)

Commentationes Mathematicae Universitatis Carolinae

For a space Z , we denote by ( Z ) , 𝒦 ( Z ) and 2 ( Z ) the hyperspaces of non-empty closed, compact, and subsets of cardinality 2 of Z , respectively, with their Vietoris topology. For spaces X and E , C p ( X , E ) is the space of continuous functions from X to E with its pointwise convergence topology. We analyze in this article when ( Z ) , 𝒦 ( Z ) and 2 ( Z ) have continuous selections for a space Z of the form C p ( X , E ) , where X is zero-dimensional and E is a strongly zero-dimensional metrizable space. We prove that C p ( X , E ) is weakly orderable if and...

Disconnectedness properties of hyperspaces

Rodrigo Hernández-Gutiérrez, Angel Tamariz-Mascarúa (2011)

Commentationes Mathematicae Universitatis Carolinae

Let X be a Hausdorff space and let be one of the hyperspaces C L ( X ) , 𝒦 ( X ) , ( X ) or n ( X ) ( n a positive integer) with the Vietoris topology. We study the following disconnectedness properties for : extremal disconnectedness, being a F ' -space, P -space or weak P -space and hereditary disconnectedness. Our main result states: if X is Hausdorff and F X is a closed subset such that (a) both F and X - F are totally disconnected, (b) the quotient X / F is hereditarily disconnected, then 𝒦 ( X ) is hereditarily disconnected. We also...

Currently displaying 41 – 60 of 240