Induced almost continuous functions on hyperspaces

Alejandro Illanes. "Induced almost continuous functions on hyperspaces." Colloquium Mathematicae 105.1 (2006): 69-76. <http://eudml.org/doc/283810>.

@article{AlejandroIllanes2006,
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 → 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.},
author = {Alejandro Illanes},
journal = {Colloquium Mathematicae},
keywords = {Almost continuous functions; Continuum; Hyperspaces; Induced functions.},
language = {eng},
number = {1},
pages = {69-76},
title = {Induced almost continuous functions on hyperspaces},
url = {http://eudml.org/doc/283810},
volume = {105},
year = {2006},
}

TY - JOUR
AU - Alejandro Illanes
TI - Induced almost continuous functions on hyperspaces
JO - Colloquium Mathematicae
PY - 2006
VL - 105
IS - 1
SP - 69
EP - 76
AB - 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.
LA - eng
KW - Almost continuous functions; Continuum; Hyperspaces; Induced functions.
UR - http://eudml.org/doc/283810
ER -