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38
Measuresofnon-compactness
1
D(B)=D({x1
,
...
,
xk−1}∪Bk)=D(Bk)<D(Ak)
.Since
k→∞
lim
D(Ak)=
0,weinferthat
D(B)=
0,andthereforetheset
B
isrelativelycompact
in
X
.Thisimpliesthatthesequence
(xn)n∈N
includesasubsequence
(xn
l)l∈N
convergentin
X
tosomepoint
x0
.Wewillprovethat
x0∈A
.
Foranarbitrary
m∈N
,letuschoose
l0∈N
insuchawaythat
nl
0>m
.
Then,
xn
l∈An
l⊆Am
for
l>l0
.Since
Am
isaclosedset,thisimplies
that
x0∈Am
.Consequently,asthepositiveinteger
m
wasarbitrary,
weinferthatx0∈A.
Since
A⊆Am
,wehave
D(A)<D(Am)
forany
m∈N
.Hence,
D(A)=
0.Thismeansthat
A
isarelativelycompactsubsetof
X
.Since
itisclosed(astheintersectionofafamilyofclosedsets),itisalso
compact.
Now,letusassumethatforany
n∈N
theset
An
isconnected,
but
A
isnot.Thismeansthatthereexisttwonon-empty,closedand
disjointsubsets
F1
and
F2
of
X
suchthat
A=F1∪F2
.Also,wemay
findtwoopenanddisjointsubsets
G1
and
G2
of
X
suchthat
F1⊆G1
andF2⊆G2(see[100,Section1.5andCorollary4.1.13]).
Wewillshowthatthereisanindex
n∈N
suchthat
An⊆G
,
where
G:=G1∪G2
.Supposeonthecontrarythatforany
n∈N
thereexistssomepoint
xn∈An\G
.(Noticethateventhoughhere
wetalkaboutthesequence
(xn)n∈N
andthepoint
x0
andusethem
inasimilarwayasinthefirstpartoftheproof,theyaredifferent
andunrelatedentities!)Inviewoftheassumptionaboutthesequence
(An)n∈N
andusingareasoningidenticaltotheoneatthebeginning
oftheproof,weinferthatthesequence
(xn)n∈N
hasasubsequence
(xn
l)l∈N
convergenttosomepoint
x0
.Itisclearthat
x0∈A\G
.Since
A⊆G
,weobtainacontradiction.Hence,weconcludethatthereexists
n∈NsuchthatAn⊆G.
As
A⊆An
,thesets
An∩G1
and
An∩G2
arenon-empty.Moreover,
theyaredisjointandopenin
An
,and
An=(An∩G1)∪(An∩G2)
.
ThismeansthatthesetAnisnotconnected–acontradiction.Hence,
thesetAisalsoconnected.
1010330Corollary
(Cantor’stheorem)
0
Let
(An)n∈N
beanon-increasing
