A reverse viewpoint on the upper semicontinuity of multivalued maps

Fenille, Marcio Colombo. "A reverse viewpoint on the upper semicontinuity of multivalued maps." Mathematica Bohemica 138.4 (2013): 415-423. <http://eudml.org/doc/260723>.

@article{Fenille2013,
abstract = {For a multivalued map $\varphi \colon Y\multimap (X,\tau )$ between topological spaces, the upper semifinite topology $\mathcal \{A\}(\tau )$ on the power set $\mathcal \{A\}(X)=\lbrace A\subset X \colon A\ne \emptyset \rbrace $ is such that $\varphi $ is upper semicontinuous if and only if it is continuous when viewed as a singlevalued map $\varphi \colon Y\rightarrow (\mathcal \{A\}(X),\mathcal \{A\}(\tau ))$. In this paper, we seek a result like this from a reverse viewpoint, namely, given a set $X$ and a topology $\Gamma $ on $\mathcal \{A\}(X)$, we consider a natural topology $\mathcal \{R\}(\Gamma )$ on $X$, constructed from $\Gamma $ satisfying $\mathcal \{R\}(\Gamma )=\tau $ if $\Gamma =\mathcal \{A\}(\tau )$, and we give necessary and sufficient conditions to the upper semicontinuity of a multivalued map $\varphi \colon Y\multimap (X,\mathcal \{R\}(\Gamma ))$ to be equivalent to the continuity of the singlevalued map $\varphi \colon Y\rightarrow (\mathcal \{A\}(X),\Gamma )$.},
author = {Fenille, Marcio Colombo},
journal = {Mathematica Bohemica},
keywords = {multivalued map; power set; upper semicontinuity; upper semifinite topology; multivalued map; power set; upper semicontinuity; upper semifinite topology},
language = {eng},
number = {4},
pages = {415-423},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A reverse viewpoint on the upper semicontinuity of multivalued maps},
url = {http://eudml.org/doc/260723},
volume = {138},
year = {2013},
}

TY - JOUR
AU - Fenille, Marcio Colombo
TI - A reverse viewpoint on the upper semicontinuity of multivalued maps
JO - Mathematica Bohemica
PY - 2013
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 138
IS - 4
SP - 415
EP - 423
AB - For a multivalued map $\varphi \colon Y\multimap (X,\tau )$ between topological spaces, the upper semifinite topology $\mathcal {A}(\tau )$ on the power set $\mathcal {A}(X)=\lbrace A\subset X \colon A\ne \emptyset \rbrace $ is such that $\varphi $ is upper semicontinuous if and only if it is continuous when viewed as a singlevalued map $\varphi \colon Y\rightarrow (\mathcal {A}(X),\mathcal {A}(\tau ))$. In this paper, we seek a result like this from a reverse viewpoint, namely, given a set $X$ and a topology $\Gamma $ on $\mathcal {A}(X)$, we consider a natural topology $\mathcal {R}(\Gamma )$ on $X$, constructed from $\Gamma $ satisfying $\mathcal {R}(\Gamma )=\tau $ if $\Gamma =\mathcal {A}(\tau )$, and we give necessary and sufficient conditions to the upper semicontinuity of a multivalued map $\varphi \colon Y\multimap (X,\mathcal {R}(\Gamma ))$ to be equivalent to the continuity of the singlevalued map $\varphi \colon Y\rightarrow (\mathcal {A}(X),\Gamma )$.
LA - eng
KW - multivalued map; power set; upper semicontinuity; upper semifinite topology; multivalued map; power set; upper semicontinuity; upper semifinite topology
UR - http://eudml.org/doc/260723
ER -