### Induced almost continuous functions on hyperspaces

Alejandro Illanes (2006)

Colloquium Mathematicae

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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]...