$IBQUFS $POWFY 4FUT -JT HPE 7POE -JOFT -JOF …sgharibian/courses/CMSC691/2016...closed Sets -Open i...
Transcript of $IBQUFS $POWFY 4FUT -JT HPE 7POE -JOFT -JOF …sgharibian/courses/CMSC691/2016...closed Sets -Open i...
21.3 Affine dimension and relative interior_
Let corn (not nee. affine).To "upgrade"C to affine set, wediefireitsaffine hull
in
affcc)=}ï 1X,+... +Okxk1×1, "-1 NiceF.oi=1}
|_
coï toeg
c
as Cc)
_ a |....|in o
C
!_ Oaffil)
ï an=1R2
ï
ï!ï ï\I..
> after is smallest affine set cont!-dining C.
_
formal:_
t affihesets SHR.I
C s...aft)(E_
ï |Affine dimensionof cis the dimension of its off CCJ.
20
ego dim is1 20 dim is2.
Interior of." Set_
'
eg | ,in _
:... --
_ JoC
Mais
"all pts strictly in ï idle\1 _ |0 \\
interior is in ifY"_ |//
Xy zO ocean"interior is empty, sheetdir unused"is
Euclidean balls -: Box, in) =x
center XEIRr¢ radiusre0 11
{ }~ - y
ego |mm
x
yïï
xof ball in 2 DBlair)
Of
11*117 y +f T
Openclosed Sets -Li nit points: =Let c E IR"
|on noo I
egg cdodo£ it in ï _ inï
rin,fro
non
intersection
Formally:*ERR is a limitptof c if
t 470'Blast)n six)) (H(C ¢in
p setdiffeenceClosed set:_
A set Cis closed -ifc contains all its limitpts.
Open set: -N setC is open if it is notclosed.
ego |on Top M satL
ï
{ 0C=34 11)ï |_
(=[0,1,Indo
enclosed set
Formally:Throw iny "
missing limit-points,getclosure of_ --
C,denoted d (C).
old Old.00 ï .. a 6O
t Inerror: _ Int (c) ={ XEC] | |_ 33¢] 870 sit.Blx,r)E if
\
ï
hï
ïFig ! ~
ï
Boundary: -zS= cks)\int(s)in throw out interior
I
_ |4 ..... in in ---e.g. in ï ï 'aadd o S=(0,1]IR
Int(s):(017)ïS= alls), Int(s)= [o, Dto, 1)=10, 1}
|Tae inTaito5=10.0),IR?Ä note!Int(s):0-
265) =elses)\!.,!..ills) = [0.1]\ of [0, D.
Relative Interioran _ ` in
: Y interior relative tooff(s)"
rebirths)= Exespero st Blair)( )_} asraff(s)
r old def of interiort "brings invision down"e.g. |in oog
recall int (5=0an:
S
affls)relint(5)= (011)
Ex 2.2 S= ExctR3/-1fx,!1,-1×21 73=0}
^ |v
27itx
y
z
1|
-
_
] !mm2affls) =intcs)=relintls)=
215) =
{XEIR3/*3=0}of{xE1R3/-1<xll1-1<x24 X3=0}Eyelike)mailhikerd|} = 1, *3=0}