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10
LechGórniewicz
NotethatanyconvexcompactsetK⊂RnoranycompactC2-manifoldwith
orwithoutboundaryisaproximativeretract.
WeshallneedalsothenotionoftheBouligandcone.LetCbeaclosed
subsetofRn.Wesaythattheset:
TC(x)={u∈Rn
|
|
|
|
liminf
h→0+
dist(x+hujC)
h
=0}
istheBouligandcontingentconetoCatx∈C.
LetO:XOYbeamultivaluedmapandf:X→Ybeasingle-valued
map.WesaythatfisaselectionofO(writtenf⊂O)iff(x)∈O(x),for
everyx∈X.
Theproblemofexistenceofgoodselectionsformultivaluedmappingsis
veryimportantinthefixedpointtheory.
Theconceptofuppersemicontinuityisrelatedtothenotionofthesmall
counterimageofopensets.Ontheotherhand,theconceptoflowersemicon-
tinuityisrelatedtothelargecounterimageofopensets.
DEFINITION2.6.AmultivaluedmapO:XOYiscalleduppersemicon-
tinuous(u.s.c.)mapifforeveryopenU⊂YthesetO11(U)isopeninX,
whereO11(U)={x∈X|O(x)⊂U}.
Intermsofclosedsets,wehave:
PROPOSITION2.7.AmultivaluedmapO:XOYisu.s.c.iffforevery
closedsetA⊂YthesetO
11
+(A)isaclosedsubsetofX.
PROPOSITION2.8.IfO:XOYisu.s.c.,thenthegraphΓϕisaclosed
subsetofX×Y.
PROPOSITION2.9.LetO:XOYbeau.s.c.mapwithcompactvalues
andletAbeacompactsubsetofX.ThenO(A)iscompact.
PROPOSITION2.10.IfO:XOYandψ:YOZaretwou.s.c.mappings
withcompactvalues,thenthecompositionψ◦O:XOZofOandψis
au.s.c.mapwithcompactvalues.
Usingthelargecounterimage,insteadofasmallone,weget:
DEFINITION2.11.LetO:XOYbeamultivaluedmap.If,foreveryopen
U⊂Y,thesetO
11
+(U)isopeninX,thenOiscalledalowersemicontinuous
(l.s.c.)map,whereO
11
+(U)={x∈X|O(x)∩U/=∅}.
