Metric Topology
http://cis.k.hosei.ac.jp/~yukita/
2
Neighborhood of a point x in 1
.on centered radiuspositive of intervalopen an contains if number
real theof odneighborho a called is subset A 1
1
xNx
N
・x-r x x+r
N
1
3
Any subset containing a neighborhood is another neighborhood.
xNNNxN
number real theof odneighborho a also is then , If.number real theof odneighborho a be Let
11
11
・x-r x x+r
N1
N1
4
Accumulation Points
.0at accumulatenot does },4,3,2,1,0{
.0at accumulatenot does },4,3,2,1{
.0at saccumulate }0,,51,
41,
31,
21,
11{
.0at saccumulate },51,
41,
31,
21,
11{
}.{ ofpoint oneleast at contains of odneighborhoeach if
at saccumulate subset A
14
13
12
11
11
A
A
A
A
xAx
xA
5
・x-r x x+r
Mm
a b ・x-r x x+r
Mm
a b
・x-r x x+r
Mm
a b ・x-r x x+r
Mm
a b
The open interval (a,b) accumulates at each a<x<b.
whatever is the case
6
・x-r x x+r
Mm
a b ・x-r x x+r
Mm
a b
・x-r x x+r
Mm
a b ・x-r x x+r
Mm
a b
The closed interval [a,b] accumulates at each axb.
whatever is the case
・x-r x=a x+r
Mm
b ・x-r x=b x+r
Mm
a
7
Derived Set
,
}0{}0,,41,
31,
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41,
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],[],[),[],(),(. of thecalled is set The
. of pointson accumulati all ofset thedenote .Let
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bababababaAA
AAA
set derived
8
Limits of Sequences
sequence. theof any tail includenot does themof None
. of odneighborho a is ),0(or ),21,
21( ),0,(Either
sequence. theoflimit anot is that show We.Let
follows. asit provecan Wediverges. ),1,1,1,1())1((
. tosaid is converge tofails that sequenceA }.|{ tailsome contains of odneighborhoeach if
, hasor , toin )( sequenceA
1
1
x
xx
kmaAxxxa
k
km
k
diverge
limitconverges
9
Limits of Sequences (Ex12,p.45)
}.1|)1({ tail thecontainswhich
)1,1( interval some contains 0 of odneighborhoEvery
0. toconverges ),41,
31,
21,
11())1((
klk
ll
k
k
k
10
1.1 Prop. A convergent sequence in 1 has a unique limit.
( () )
Suppose we have two limits x and y. We can separate them by some of their neighbors as shown below.
ion.contradict a shows which , have we therefore time,same at the and have We
sequence. same theof taila also is . ilanother ta and tailsomeThen
21
21
JITJTIT
TTTJTIT
x y
I J
11
1.2 Monotonic Limits Theorem
numbers. real ofset theoffeature theis boundlower greatest or boundupper least of existence The
limit. a as }1|glb{ has )( sequence decreasing boundedA (b)limit. a as }1|lub{ has )( sequence increasing boundedA (a)
kaxakaxa
kk
kk
.3222
122
1 21
11
11
4321
321
!21
!11
!01
because converges !
10
e
ne
n
12
Cauchy sequence
Why?sequence.Cauchy a isit then ,convergent is )( If
.length of intervalan in contained is
sequence theof tailsome ,0number realeach for if, in sequenceCauchy a is )( sequenceA 1
k
k
a
rr
a
Remark.
13
1.3 Convergence Characterization
.in limit no hasit However, .in sequence
Cauchy a is )( sequence The .10
210Let
.in sequenceCauchy isit
ifonly and if in sequence convergent a is sequnceA 1
1
kk
k
k aa
Remark.
14
Accumulation and Convergence
points.on accumulati no hasbut ,convergentclearly isconstant eventually is that )( sequenceA 15 Ex.
.1 and 1 pointson accumulati twohasIt
diverges. )11()1( sequence The Ex.14
k
k
a
k
15
1.4 Limit-Accumulation Properties
To be filled in the future.
16
・x ・x ・xr
rr
Open n-ball about x with radius r
17
・x ・x ・xr
rr
Closed n-ball about x with radius r
18
・x ・x ・xr
rrN
N N
Neighborhood in n of a point x
19
2.1 Neighborhood property
points. its ofeach of odneighborho a is ball- closed a of complement the(b)
and points, its ofeach of odneighborho a is ball-open an (a), space-Euclidean In
nnn n
20
Open set
.),( ball-open an is there
,point each for if, in open is subset A n
nn
rxBn
UxU
21
Closed set
. ofpoint on accumulatian is sayscondition theOtherwise satisfied. iscondition the, if fact,In
.in contained is intersects of nbhdevery such that any if
in closed is say that can wely,Equivalent
points.on accumulati its all contains if in closed is EFsubset A n
FxFxF
Fxx
F
F
n
nn
n
Remark.
22
Propositions
1. A subset is open in n if and only if its complement is closed in n.
2. Any union of open sets is open.3. Any intersection of closed sets is closed.
23
A subset is open in n if and only if its complement is closed in n.
.in is intersects of nbhdevery whosepoint Any
. intersectsnot does that nbhd a has point Any
nbhd a has point Any . and Let
FFxx
FFx
UUxUFFU
n
n
nn
UF
24
Any union of open sets is open.
rxU
UrxB
UrxBrUUxT
UxU
T
T
TT
)llRadius(getInnerBa.
.),( have weThus
.),( have we0 somefor open, is Since.such that be Let
. and setsopen ofunion a be Let
Remark.
25
Any intersection of closed sets is closed.The dual of the previous proposition
. means Thisclosed. is since have weany for thusand
,any for on with intersecti has nbhd theClearly,.on with intersecti has of nbhdevery such that Let
sets. closed ofon intersecti a be Let
.closedness of definitionour fromdirectly edemonstrat usLet
FxFFxT
TFFxx
FF
n
T
26
An open set is a union of open balls.
.),( have weClearly,
.),(ballopen an is there,any For set.open an be Let
UrxB
UrxBUxU
Uxx
x
27
Metric subspaces
).,(in close is where,),(in closed is (b)
).,(in open is where,),(in open is (a)
:Then ).,( of subspace metric a be ),(Let
dYYFFXGdXXG
dYYUUXVdXXVdYdX
28
Theorem 3.5
. and such that setsopen disjoint exist There.in sets closeddisjoint be and Let )(Normality (b)
.in odsneighborhodisjoint have and pointsDistinct property) (Hausdorff (a)
:properties twohas ),( space metricEvery
VBUAXBA
Xyx
dX
Notice that (a) is a special case of (b).
29
Proof of Th. 3.5(b)
. and
of choice the theocontrary t ),2
,(or )2
,(either So,
).,max(22
),(),(),( have weThen,
., somefor )2
,()2
,(
is e that thersuppose we, see To., have weandopen are and Clearly,
.)2
,( and )2
,( Take
. and closed is since ),(;. and closed is since ),(;
ba
ab
baba
ba
Bb
b
Aa
a
b
a
rr
raBbrbBa
rrrrxbdxadbad
BbAarbBraBx
VUVBUAVU
rbBVraBU
AbAAXrbBAbBaBBXraBAa
30
Closure
• Omitted
31
Continuity
)).((CLS)( implies )(CLS,point each and
,each for that,provided continuous is :function A spaces. metric ofpair any be ),( and ),(Let
AfxfAxXx
XAYXfdYdX
YX
YX
fA f(A)xf(x)
This kind of situation violates the condition.
32
Pinching is continuous.
・・
2:p
33
Gluing is continuous
34
4.1 Continuity Characterization
).()(limlim);( (g).))(),((),( s.t. 0;0; (f)
open. )(;open (e)
closed. )(; closed (d)
)).((CLS))((CLS; (c)
)).((CLS))(CLS(; (b).continuous is function The (a)
:),( and ),( spaces metricsbetween :function afor conditions equivalent are following The
1
1
11
xfafxaaszfxfdrzxdrsXx
XVfYV
XFfYF
BfBfYB
AfAfXAf
dYdXYXf
kkkkk
YX
YX
YX
YX