Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Colloquium Mathematicum / All issues

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.