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 → Y be an almost continuous function. Let C(f): C(X) → C(Y) and $2^f:2^X\to 2^Y$ be the induced functions given by $C\left(f\right)\left(A\right)=c{l}_Y\left(f\left(A\right)\right)$ and $2^f\left(A\right)=c{l}_Y\left(f\left(A\right)\right)$. In this paper, we prove that: • If $2^f$ is almost continuous, then f is continuous. • If C(f) is almost continuous and X is locally connected, then f is continuous. • If X is not locally connected, then there exists an almost continuous function f: X → [0,1] such that...

Given a metric continuum $X$ and a positive integer $n$, $F_n\left(X\right)$ denotes the hyperspace of all nonempty subsets of $X$ with at most $n$ points endowed with the Hausdorff metric. For $K\in F_n\left(X\right)$, $F_n(K,X)$ denotes the set of elements of $F_n\left(X\right)$ containing $K$ and $F_n^K\left(X\right)$ denotes the quotient space obtained from $F_n\left(X\right)$ by shrinking $F_n(K,X)$ to one point set. Given a map $f:X\to Y$ between continua, $f_n:F_n\left(X\right)\to F_n\left(Y\right)$ denotes the induced map defined by $f_n\left(A\right)=f\left(A\right)$. Let $K\in F_n\left(X\right)$, we shall consider the induced map in the natural way $f_{n,K}:F_n^K\left(X\right)\to F_n^{f\left(K\right)}\left(Y\right)$. In this paper we consider the maps $f$, $f_n$, $f_{n,K}$ for some $K\in F_n\left(X\right)$ and $f_{n,K}$ for...

We construct examples of mappings $f$ and $g$ between locally connected continua such that $2^f$ and $C\left(f\right)$ are near-homeomorphisms while $f$ is not, and $2^g$ is a near-homeomorphism, while $g$ and $C\left(g\right)$ are not. Similar examples for refinable mappings are constructed.