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Final

# MAT337 Entire Class Lecture Notes

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School
University of Toronto St. George
Department
Mathematics
Course
MAT337H1
Professor
I.M Sigal
Semester
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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