Induced almost continuous functions on hyperspaces
Volume 105 / 2006
Colloquium Mathematicum 105 (2006), 69-76
MSC: Primary 54B20.
DOI: 10.4064/cm105-1-8
Abstract
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.
