Metric Topology

http://cis.k.hosei.ac.jp/~yukita/

2

Neighborhood of a point x in 1

.on centered radius

positive of intervalopen an contains if number

real theof odneighborho a called is subset A 1

1

x

Nx

N

･x-r x x+r

N

1

3

Any subset containing a neighborhood is another neighborhood.

xNNN

xN

number real theof odneighborho a also is then , If

.number real theof odneighborho a be Let

11

11

･x-r x x+r

N1

N

１

4

Accumulation Points

.0at accumulatenot does },4,3,2,1,0{

.0at accumulatenot does },4,3,2,1{

.0at saccumulate }0,,5

1,

4

1,

3

1,

2

1,

1

1{

.0at saccumulate },5

1,

4

1,

3

1,

2

1,

1

1{

}.{ of

point oneleast at contains of odneighborhoeach if

at saccumulate subset A

14

13

12

11

11

A

A

A

A

xA

x

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,,4

1,

3

1,

2

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1

1{},

4

1,

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],[],[),[],(),(

. of thecalled is set The

. of pointson accumulati all ofset thedenote

.Let

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1

bababababa

AA

AA

A

set derived

8

Limits of Sequences

sequence. theof any tail includenot does themof None

. of odneighborho a is ),0(or ),2

1,

2

1( ),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

kmaAx

xxa

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 ),4

1,

3

1,

2

1,

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1()

)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

JIT

JTIT

TTT

JTIT

x y

I J

11

1.2 Monotonic Limits Theorem

numbers. real ofset theof

feature 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)

kaxa

kaxa

kk

kk

.3222

1

22

1

2

1

1

1

1

1

432

1

32

1

!2

1

!1

1

!0

1

because converges !

1

0

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

r

r

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 ,convergent

clearly isconstant eventually is that )( sequenceA 15 Ex.

.1 and 1 pointson accumulati twohasIt

diverges. )1

1()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

n

n

n 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 says

condition 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

Fx

Fx

F

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

UUx

UFFU

n

n

nn

UF

24

Any union of open sets is open.

rxU

UrxB

UrxBrU

UxT

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 This

closed. 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

Fx

FFxT

TF

Fxx

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

UrxB

UxU

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

dYYF

FXGdXXG

dYYU

UXVdXXV

dYdX

28

Theorem 3.5

. and such that setsopen disjoint exist There

.in sets closeddisjoint be and Let )(Normality (b)

.in odsneighborho

disjoint have and pointsDistinct property) (Hausdorff (a)

:properties twohas ),( space metricEvery

VBUA

XBA

X

yx

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

raBb

rbBa

rrrr

xbdxadbad

BbAar

bBr

aBx

VU

VBUAVU

rbBV

raBU

AbAAXrbBAb

BaBBXraBAa

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

AfxfAx

Xx

XAYXf

dYdX

YX

YX

fA f(A)x

f(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

xfafxaa

szfxfdrzxdrsXx

XVfYV

XFfYF

BfBfYB

AfAfXA

f

dYdX

YXf

kk

kk

k

YX

YX

YX

YX