122
Section5.1
Structureof
R
n
289
Listraversedinoppositedirectionsastandvaryfrom1to1in(5.1.11)and(5.1.13).
Settingt D1in(5.1.11),weseethatX
1
D.1;1;1/isalsoonL.Thelinesegment
fromX
0
toX
1
consistsofallpointsoftheform
XDt.1;1;1/C.1t/.1;3;1/; 0t1:
ThesefamiliarnotionscanbegeneralizedtoR
n
,asfollows:
Definition5.1.10
SupposethatX
0
andUareinR
n
andU¤0.Thenthelinethrough
X
0
inthedirectionofUisthesetofallpointsinR
n
oftheform
XDX
0
CtU; 1<t<1:
Asetofpointsoftheform
XDX
0
CtU; t
1
tt
2
;
iscalledalinesegment.Inparticular,thelinesegmentfromX
0
toX
1
isthesetofpointsof
theform
XDX
0
Ct.X
1
X
0
/DtX
1
C.1t/X
0
; 0t1:
NeighborhoodsandOpenSets in
R
R
R
n
HavingdefineddistanceinR
n
,wearenowabletosaywhatwemeanbyaneighborhood
ofapointinR
n
.
Definition5.1.11
If>0,the-neighborhoodofapointX
0
inR
n
istheset
N
.X
0
/jD
˚
X
ˇ
ˇ
jXX
0
j<
:
An-neighborhoodofapointX
0
inR2 istheinside,butnotthecircumference,ofthe
circleofradiusaboutX
0
.InR3itistheinside,butnotthesurface,ofthesphereofradius
aboutX
0
.
InSection1.3westatedseveralotherdefinitionsintermsof-neighborhoods:neigh-
borhood,interiorpoint,interiorofaset,openset,closedset,limitpoint,boundarypoint,
boundaryofaset,closureofaset,isolatedpoint,exteriorpoint,andexteriorofaset.Since
thesedefinitionsarethesameforR
n
asforR,wewillnotrepeatthem.Weadviseyouto
readthemagaininSection1.3,substitutingR
n
forRandX
0
forx
0
.
Example5.1.6
LetS bethesetofpointsinR2 inthesquareboundedbythelines
x D ˙1,y D D ˙1, , exceptfortheoriginandthepointsontheverticallinesx D D ˙1
(Figure5.1.4,page290);thus,
SD
˚
.x;y/
ˇ
ˇ
.x;y/¤.0;0/;1<x<1;1y1
: