Induced almost continuous functions on hyperspaces

Alejandro Illanes

Colloquium Mathematicae, Tome 106 (2006), p. 69-76 / Harvested from The Polish Digital Mathematics Library

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 → Y be an almost continuous function. Let C(f): C(X) → C(Y) and $2^{f}: 2^{X} → 2^{Y}$ be the induced functions given by $C(f)(A) = cl_{Y}(f(A))$ and $2^{f}(A) = cl_{Y}(f(A))$. 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 C(f) is almost continuous and f is not continuous.

Détail

Publié le : 2006-01-01

Zbl 1102.54008

EUDML-ID : urn:eudml:doc:283810