Induced almost continuous functions on hyperspaces

Alejandro Illanes¹

¹ Instituto de Matematicas UNAM Circuito Exterior, Ciudad Universitaria 04510 México, D.F., Mexico

Colloquium Mathematicum, Tome 105 (2006) no. 1, pp. 69-76.

Résumé

For a metric continuum $X$, let $C(X)$ (resp., $2^{X}$) be the hyperspace of subcontinua (resp., nonempty closed subsets) of $X$. Let $f:X\rightarrow Y$ be an almost continuous function. Let $C(f):C(X)\rightarrow C(Y)$ and $ 2^{f}:2^{X}\rightarrow 2^{Y}$ be the induced functions given by $C(f)(A)=$ ${\rm cl}_{Y}(f(A))$ and $2^{f}(A)={\rm cl}_{Y}(f(A))$. In this paper, we prove that:$\bullet$ If $2^{f}$ is almost continuous, then $f$ is continuous. $\bullet$ If $C(f)$ is almost continuous and $X$ is locally connected, then $f$ is continuous.$\bullet$ If $X$ is not locally connected, then there exists an almost continuous function $f:X\rightarrow \lbrack 0,1]$ such that $C(f)$ is almost continuous and $f$ is not continuous.

Détail

DOI : 10.4064/cm105-1-8

Keywords: metric continuum resp hyperspace subcontinua resp nonempty closed subsets rightarrow almost continuous function rightarrow rightarrow induced functions given paper prove bullet almost continuous continuous bullet almost continuous locally connected continuous bullet locally connected there exists almost continuous function rightarrow lbrack almost continuous continuous