Lecture 3

Pset I has been posted on the course webpageandMoodleDue Date 27 042021 at 3 15PM

Problem sessiontoday no open officehour

Recall metric SpaceX

Openbails opensets neighbourhoods

convergenceof8equence in metricspacexcontinuousfunction

ydhomeomorphism

Uopen f U isopen f continuousbijection

x ft continuous

Lemme Rn de E Brio de

Corr All openbails in 4127de are homeomorphic

to each other and to IR de

foo Br107 0 112 bijective

n o X_ continuousy 11 112

f g IR Brio continuous

y no

I HYHz r

HDetJn Xie a subset A ex is said to be

boundedg F M ZO est

d aib TY tf dis EA

Belo DE is bounded Khidr is NOT

bounded

1122

Deff A topological property is the one which is

preserved by homeomorphism

Boundedness is NOT a topological property

X d and X d

DefI Two metrics domd d on X one called

topologically equivalent if id X d Xcel1

is a homeomorphism ni se

I

Exeir id Xcel x d is a homeomorphism

o_0

Nn a ire Xcel 0 8 Xn X lie X d

0 0

opensets in Xcel 0 0 open set in x d's

Compactness

Suppose I is an index set

A collection Vaz of open sets of X Uac X

fats is an open cover of A EX ifA e U K

LEI

Remarke This def makes sense for arbitrary

topological spaces

o 44 r

Defn A subset ACH d is compactiy either oftheequivalent condition holds

a Every open cover Valdez of A has a

finite subcorer i e there is a finite subset

Li 22 Ln E I b fn

A e U ki i

iz I

Every sequence rn c A has a convergent

subsequencew limit in A

She 1121

is compact in 112

theorem Compactness is a topologicalproperty i e

f X y is continuous and suppose A e X

is compact then so is fCA EY

Proof Ac X is compact

Want f A E'S is compact

het Va zbe an open cover for fCA

f A e UVaAFI

Look at f Va open aie X as f is continuous

f Va Idf is an open cover forA

F f 2 Oz dn f c Is.tnA C U f Vai

in lM

f A e U Vai5 1

Topological spaces

def X is a metric fortopological spacea interior of a subset A X is the set

AO ne Al F a nbdUofX ai X w

U e A

not a clusterpoint

b closure of A exis the set

A xe X every nbd of xuiX intersects AEsee It is called a cluster point

Exes A C X

Ao is the largest open subsetofX that is

contained in A i e

Ao U UU ex open and UCA

F is the smallest closed subset of X thatcontains A ii e

A A V

EX closed USA

Note Arbitrary union of open sets is open ina metric space

KCI Ux D XE Uxo F some Br Cyoopen

Brad e U kD a I

Arbitrary intersectionof closed sets is dosed

D Ii p

f th't oR

f th th of isclosed in IR

111 f I l l l il

Def A topology on a set X is a collection T

ofsubsets ofX satisfying the followingi ol X e T

ii if Ua IET 12 C T

iii if Vail c I I Va ET

Def Elements of T subsetsof X are calledOpensets in X

O x x o