Induced mappings on hyperspaces $F_n^K(X)$

Castañeda-Alvarado, Enrique ; Mondragón-Alvarez, Roberto C. ; Ordoñez, Norberto

Commentationes Mathematicae Universitatis Carolinae, Tome 65 (2024) no. 1, pp. 79-97

Abstract (VO)

Given a metric continuum $X$ and a positive integer $n$, $F_{n}(X)$ denotes the hyperspace of all nonempty subsets of $X$ with at most $n$ points endowed with the Hausdorff metric. For $K\in F_{n}(X)$, $F_{n}(K,X)$ denotes the set of elements of $F_{n}(X)$ containing $K$ and $F_{n}^K(X)$ denotes the quotient space obtained from $F_{n}(X)$ by shrinking $F_{n}(K,X)$ to one point set. Given a map $f\colon X\to Y$ between continua, $f_{n}\colon F_{n}(X)\to F_{n}(Y)$ denotes the induced map defined by $f_{n}(A)=\nobreak f(A)$. Let $K\in F_{n}(X)$, we shall consider the induced map in the natural way $f_{n,K}\colon F_{n}^K(X)\to F_{n}^{f(K)}(Y)$. In this paper we consider the maps $f$, $f_{n}$, $f_{n,K}$ for some $K\in F_n(X)$ and $f_{n,K}$ for each $K\in F_n(X)$; and we study relationship between them for the following classes of maps: homeomorphisms, monotone, confluent, light and open maps.

Détail

DOI : 10.14712/1213-7243.2024.016

Classification : 54B15, 54B20, 54C05, 54C10
Keywords: continuum; symmetric product; quotient space; hyperspace; induced mapping