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I.M Sigal
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Final

# MAT337 Entire Class Lecture Notes

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University of Toronto St. George

Mathematics

MAT337H1

I.M Sigal

Winter

Description

NOTES
General Le Continu
Topology
maximum
MA1320
Hand out
Today
all tuctions ter 1 sections
chapter section
funch
continuous
st be defied -t.
NOTES
ceo whenever UCR open
Def UCR.
'open
ex Well-behaved
n U
set
NOTES General Le Continu Topology maximum MA1320 Hand out Today all tuctions ter 1 sections chapter section funch continuous st be defied -t. NOTES ceo whenever UCR open Def UCR. 'open ex Well-behaved n U setNOT
01 CM 01
contivuuous,
Lee UCR be.
then
be
Pen D
EU Such exask because is
the
Coutinul
then R -f
S
EU
f
We nees a
second half of
do later
NOTES
Care.
A space
is amtli catisfy
Court
set
FI(U) is open in R
oper sets
Unions f open sek open
mes secti
are Open
volk.
NOT 01 CM 01 contivuuous, Lee UCR be. then be Pen D EU Such exask because is the Coutinul then R -f S EU f We nees a second half of do later NOTES Care. A space is amtli catisfy Court set FI(U) is open in R oper sets Unions f open sek open mes secti are Open volk.NOTES
et. A Let be.
Pefiniti
to
llectisn
on X
St
Len
X, for
TA under strod
Hauty f*ed or, we call the sets in "open
uel spas
Conti
and f 9 Cont
then
is cont.
OTES
Need to
(V) E
C lai
is in
Kramples
X (R
open in the std. Seusa
2, X
NOTES et. A Let be. Pefiniti to llectisn on X St Len X, for TA under strod Hauty f*ed or, we call the sets in "open uel spas Conti and f 9 Cont then is cont. OTES Need to (V) E C lai is in Kramples X (R open in the std. Seusa 2, Xfinite
NOTES
f: IR
(Rtnu
Sup
B, UB
NOTES
finite as
since ts finite
court.
ullen it
in the US
cont.
constan
ren
std
non-em
std
fram week 3
finite NOTES f: IR (Rtnu Sup B, UB NOTES finite as since ts finite court. ullen it in the US cont. constan ren std non-em std fram week 3NOTES
take B
balls.
a, b
b EIR.
Lt Lauwer limit basis
relati
Set
True false
One
sifu
the diti manu under
NOTE
a, b
Ufa,
an is the minimum
b is the mass
1 the smallest topo
they collect
of sets
in B
NOTES take B balls. a, b b EIR. Lt Lauwer limit basis relati Set True false One sifu the diti manu under NOTE a, b Ufa, an is the minimum b is the mass 1 the smallest topo they collect of sets in BRead Along
NOTES
closed under
five
than
cochere
weaken
st.
all, even
un
C 2
ntesse eturn
Gwen
a basis
l The smallest
Kit
Then OT
Let d
is a
it is the smallest t
Heat contains
terse
n. clearly contains 8, it
is
contains
finite intersections ind tutum
EB, CU
3 B,
Read Along NOTES closed under five than cochere weaken st. all, even un C 2 ntesse eturn Gwen a basis l The smallest Kit Then OT Let d is a it is the smallest t Heat contains terse n. clearly contains 8, it is contains finite intersections ind tutum EB, CU 3 B,ttanol
proof of
can
Enough that
them
True as GJT
(a
basis
a
-the
than -the-
gn R R
a, b
U L
atn,
b y
but war in
ed
Exercise.
Comt.
3 Show
that converses
an fets
NOTES
ttanol proof of can Enough that them True as GJT (a basis a -the than -the- gn R R a, b U L atn, b y but war in ed Exercise. Comt. 3 Show that converses an fets NOTESNOT
goof
taken xe IR let co b y be
boss
contains
tandane topology
at
b
St
We showed that
L is timer than
which
-tak
basis elment that contains
boris
We Can And
lower
Satria
On We
amy basis
A, b) e
Carb
to
NOTES
Lect
Produ uk
spaces Want
SA.
band.
We w
be
Theorem
I & 7.
NOT goof taken xe IR let co b y be boss contains tandane topology at b St We showed that L is timer than which -tak basis elment that contains boris We Can And lower Satria On We amy basis A, b) e Carb to NOTES Lect Produ uk spaces Want SA. band. We w be Theorem I & 7.NOTES
a
restlichun at stage n
at
have
E A
d (X
these are satisfied na
2.2
EA
than n
Clar
s Cauchy
nk
So
utions
are nowhere
T Wednesday
2. the fnal
3 Hilbert
and has
thin
terr
CUde.
A Boing. Space
is a space X suuh thap
Dom collecti
JM
X has
a unism hay no
mber or
in
open ses interseds the
terror
churn
m A complete
a Parte space
be many
Let Fi
closes seh
navir n in error
and
U- FI is open
NOTES a restlichun at stage n at have E A d (X these are satisfied na 2.2 EA than n Clar s Cauchy nk So utions are nowhere T Wednesday 2. the fnal 3 Hilbert and has thin terr CUde. A Boing. Space is a space X suuh thap Dom collecti JM X has a unism hay no mber or in open ses interseds the terror churn m A complete a Parte space be many Let Fi closes seh navir n in error and U- FI is openNOTES
indudiĆ©n, find a
2 B, n, rn
be the
any n
emphy
ed
batre G Weny Coun Collection spen
a devise
dense
erect of
on Ware spa ce
a C
dense set,
bia
is Com?
and Poire. Mit .tho set
s bi
t has
E
exercise
a P
Sith
dense
Un
Un dense
rtati
NOTES indudiĆ©n, find a 2 B, n, rn be the any n emphy ed batre G Weny Coun Collection spen a devise dense erect of on Ware spa ce a C dense set, bia is Com? and Poire. Mit .tho set s bi t has E exercise a P Sith dense Un Un dense rtatiNOTES
If U
Pan Y
are.
Calm
S is a basis Rr
be the topdyry
cont.
h check
were
of
front
NOTES
a subset
poduct
as,
two.
a
B.
as a Sub
the
ten fre
MSI j
bur not in the
30 B
deed
is spen
open
NOTES If U Pan Y are. Calm S is a basis Rr be the topdyry cont. h check were of front NOTES a subset poduct as, two. a B. as a Sub the ten fre MSI j bur not in the 30 B deed is spen openNOT
A an interval
relaki 2 The order, them two
On
A set F in a
Ufa
FE
are
Ark it
p(92 is closed B closed
ed in
closed a
St. Ba TAF
closed
BC
Lee S
See
A Rea
along Marlon
answer general-interest zmuest anc Pa3nd
by email
these
C Reminders
Oasis for
3 f closed Gs FC open.
inherited from
open
A set
A b
A
Example The Carter Set C:
d.52d
NOT A an interval relaki 2 The order, them two On A set F in a Ufa FE are Ark it p(92 is closed B closed ed in closed a St. Ba TAF closed BC Lee S See A Rea along Marlon answer general-interest zmuest anc Pa3nd by email these C Reminders Oasis for 3 f closed Gs FC open. inherited from open A set A b A Example The Carter Set C: d.52dNOT
dole of
the open
VMI
removed: C
Merval
bid on
C:
1: L d
CC
The ter
expas
of x hau
no
an un C
Set
of DR2
Cun
St.
NOTE
a subset
ast cloyed set
A)
catami
A Cl A
Closure of A
Irrtersecten f all closed sok
A
f A
A
Pen sets in A
20 A
intersects A
Let B b
nbol
but B
so A
if A than A open
A CA
A D A
so ACAA
a nbd
So A
NOT dole of the open VMI removed: C Merval bid on C: 1: L d CC The ter expas of x hau no an un C Set of DR2 Cun St. NOTE a subset ast cloyed set A) catami A Cl A Closure of A Irrtersecten f all closed sok A f A A Pen sets in A 20 A intersects A Let B b nbol but B so A if A than A open A CA A D A so ACAA a nbd So ANOTES
Class photo 2.ss
at hart of TA
kiwn is due at 4:00
will b
on eh
Munkres T-17
Read Al
On
ty A
this
UK.
n K
k open
st.
A
Bd A A A
Exam
A MA
VOTES
C (a
Az AUA
A SA
A A
A
int closed sef
closed hut SL
isnt.
A space s called
TI space fi
pint in t is clased
COM
there is an open 3d- KE
NOTES Class photo 2.ss at hart of TA kiwn is due at 4:00 will b on eh Munkres T-17 Read Al On ty A this UK. n K k open st. A Bd A A A Exam A MA VOTES C (a Az AUA A SA A A A int closed sef closed hut SL isnt. A space s called TI space fi pint in t is clased COM there is an open 3d- KENOTES
Pen sci
co M IS Close
closed
ale
a ra reation axiom
st.
t (not
ev.
but
closed
check State Point sea,
because
campume, d
To
NOTES
Haus dort
zkeU
UAV
fe V
closed set F
F a U, V open
cu
fi panies f Hausdorff (Tal
paces
Anu is hire
2, a
has
mart one f in
produts Tr
Stubs pales
NOTES Pen sci co M IS Close closed ale a ra reation axiom st. t (not ev. but closed check State Point sea, because campume, d To NOTES Haus dort zkeU UAV fe V closed set F F a U, V open cu fi panies f Hausdorff (Tal paces Anu is hire 2, a has mart one f in produts Tr Stubs palesNOTES
of uf 2.
A
is close
is close
all in Anu
distinct
By induction
aN
Sit
if i N. then
Txi EU
aivun (ti) suppose Y
are bor
bd of Y
a n
nite man
Con
wise
like
NOTES
icture
T TT T 32,T
NOTES of uf 2. A is close is close all in Anu distinct By induction aN Sit if i N. then Txi EU aivun (ti) suppose Y are bor bd of Y a n nite man Con wise like NOTES icture T TT T 32,TNOTES
Cont. TFAE
Court.
nbd
a nbd
open
E un fCA)
(A) U.
NO
res Ibi
TPRE
s Cont
Closed)
2-3
Let BC be closed
Let ft (B) A
ANTS A
is closel
ACA Yet ACA
A A
So A
is c
The p
S. far each 2 e T
Pose
NOTES Cont. TFAE Court. nbd a nbd open E un fCA) (A) U. NO res Ibi TPRE s Cont Closed) 2-3 Let BC be closed Let ft (B) A ANTS A is closel ACA Yet ACA A A So A is c The p S. far each 2 e T PoseNOTES
x, E X
ramp
set of sequences
Then T -t
If
St. fra
Example
Comsetuence of AC
The box
NOTES
Possibility Z
or a
We want
conti
F(z)- (e to be
Read Along
TTU
seek
Coht
csnt
NOTES x, E X ramp set of sequences Then T -t If St. fra Example Comsetuence of AC The box NOTES Possibility Z or a We want conti F(z)- (e to be Read Along TTU seek Coht csntNOTES
open should
a
Claim
2
Seda sfies
cel Hence
Ani
ope
Ula
then
NOTES
Cla
cla
it is finite
t
t, t
dear
Conte
in tur
f is, comit
the tw
TTAJ are the sane
X2 is
X is h
TTAo.
NOTES open should a Claim 2 Seda sfies cel Hence Ani ope Ula then NOTES Cla cla it is finite t t, t dear Conte in tur f is, comit the tw TTAJ are the sane X2 is X is h TTAo.a
a
pact of 3
pick.
lake.
bex
Defi
A a
the
metric
2 IR
Consid
are
where U
ETTA
3 e
tale y e UMTA
t's more 'inefric thou
metric space
NOTES
a a pact of 3 pick. lake. bex Defi A a the metric 2 IR Consid are where U ETTA 3 e tale y e UMTA t's more 'inefric thou metric space NOTESBiv
ball mod
a abas
dien
Br (xo
me nizable
Th
metric d
in
a ues
the of X
Max
NOTES
Ad
Red 20, 21, 23.
2
d CP, ri
en
v nbo f of TX, all but finitely
the X
F's are in
metric
NOTES
Biv ball mod a abas dien Br (xo me nizable Th metric d in a ues the of X Max NOTES Ad Red 20, 21, 23. 2 d CP, ri en v nbo f of TX, all but finitely the X F's are in metric NOTESDATE
Clea
vmetr.
LA
r a
Be
diode
Crain
net metri able
at
Clam
O E
NOTES
let u be a b
the
So UAA
E A
St.
chosen O o o o
air isnur
NOTES
DATE Clea vmetr. LA r a Be diode Crain net metri able at Clam O E NOTES let u be a b the So UAA E A St. chosen O o o o air isnur NOTESNOTES
as
Let U
Gas
spews.
So ak
12 box
G etruzable.
k is inetrizalola
EX is constant
CA
to say
Lamma rs mein zable t has
a metric
etrt c
Cam
NOT
s Let be
d is a
vned TCC
AamA
20
NOTES as Let U Gas spews. So ak 12 box G etruzable. k is inetrizalola EX is constant CA to say Lamma rs mein zable t has a metric etrt c Cam NOT s Let be d is a vned TCC AamA 20Cla
NOTES
1
cla...,
NOTES
Choose -ma
te m
A
A (A
Places
13 dens
ter sects
te sa
Cla NOTES 1 cla..., NOTES Choose -ma te m A A (A Places 13 dens ter sects te saNOTES
De AC
clause if A
are den
nable it t has
a cova
Sepa
dense
s
l, IR is rable
St. Pa
AC B Sr
A po eith
hall to
front
NOTES
Read
Munkres 23
24
a
The Trtermediate value theorem
s connected
foto
both
ark
Example
Equivalent
F. aosas and non-ent'ty
t is Tooth
NOTES De AC clause if A are den nable it t has a cova Sepa dense s l, IR is rable St. Pa AC B Sr A po eith hall to front NOTES Read Munkres 23 24 a The Trtermediate value theorem s connected foto both ark Example Equivalent F. aosas and non-ent'ty t is ToothNOT
s Commeded
has
separation
Assume Then
is not
Connected
dopen.
IS COMnect2
NTS A
males Seese
CA
not
OEG
aim
closes. EA
S
A
S, CI-S, IT c A
bud then
mbalst
this an open
connected
Sup
subset f of IR
s either
an
er Sem
open or a
ray.
or the
wh ole th
or the
empty S
A conveys
then A
(2,too
A (n
NOT s Commeded has separation Assume Then is not Connected dopen. IS COMnect2 NTS A males Seese CA not OEG aim closes. EA S A S, CI-S, IT c A bud then mbalst this an open connected Sup subset f of IR s either an er Sem open or a ray. or the wh ole th or the empty S A conveys then A (2,too A (nNOTES
A net connecte
Co, 3
Suppose C CUA
dopan
an nar- empty
CA-AD.
s connected CSA
Avery clopen set in K
Connectaa
ected is
Convex
intense ser
nected
s-G problems
ods) Theorems from
class
NOTES
Care Comm
teorem
then so is
and
is
Cross
ted, be
two
nected set.
witti a non-trivial intersection
So f
Cenneesta
Cunnecte.
NOTES A net connecte Co, 3 Suppose C CUA dopan an nar- empty CA-AD. s connected CSA Avery clopen set in K Connectaa ected is Convex intense ser nected s-G problems ods) Theorems from class NOTES Care Comm teorem then so is and is Cross ted, be two nected set. witti a non-trivial intersection So f Cenneesta Cunnecte.continuous, then
IS als

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