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# GENERAL TOPOLOGY AND FUNDAMENTAL GROUPS MTH 631

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This 34 page Class Notes was uploaded by Miss Johan Jacobson on Monday October 19, 2015. The Class Notes belongs to MTH 631 at Oregon State University taught by D. Garity in Fall. Since its upload, it has received 16 views. For similar materials see /class/224440/mth-631-oregon-state-university in Mathematics (M) at Oregon State University.

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Countable Dense sets Read Section 14 Def A subset D of a topological space X is dense in X if each nonempty open set in X contains a point of D Skim sections 9 and 10 Examples Continue to work on homework Theorem 0 If X has a countable basis then X has a countable dense subset 9 If X is metrizable and has a countable dense subset then X has a countable basis Cor Hg is not metrizable Mm est rFallZOOE Order Topology lS Mm est rFallZOOE Order Topology 23 Order Topology Def A simple or linear order on X is a transitive relation Def If X is linearly ordered with more than one element lt on X such that V pair X y in X exactly one of the the order topology on X is the topology with basis following holds consisting of all intervals of one of the forms Xlty o abforaltb y lt X o m b where m is the minimum element of X if one exists X y39 o a M where M is the maximum element of X if one exists Check this collection of intervals is a basis for some topology using the results of Section 13 Mm est rFallZOOE Order Topology 33 Mm est rFallZOOE Order Topology 43 Examples 0 H usual order 0 Ix l dictionary order 0 2 well ordered set Def A linearly ordered set is well ordered it every nonempty subset has a smallest element Well Ordering Well Ordering Theorem Every set can be well ordered Note this is equivalent to the axiom of choice Section 9 Def If X is linearly ordered and 0c 6 X 80 XX lt 0c is called the section of X by or Mm eal rFall zoos Order Topology 53 Mm eal rFall zoos Order Topology era Cor There exists an uncountable well ordered set each section of which is countable Proof of Corollary If X is well ordered uncountable with a maximal element and if Q is the smallest element with uncountably many predecessors SQ is the desired example Basic Examples 0 H usual metric and usual topology 0 Metric Spaces 0 Hg 0 X l dictionary order topology 0 SQ 0 E R H 71 I And subspaces products and quotients of these Mm eal rFall zoos Order Topology Mm esl rFall zoos Order Topology Major Theorems Embedding Theorem Every separable metric space is homeomorphic to a subspace of l E 01 Tychonoff s Theorem An arbitrary product of compact spaces is compact Urysohn s Lemma If A B are closed and disjoint in a normal space X El a continuous function f X gt 01 such that fA 0 and fB Other Major Theorems Tietze s Extension Theorem lf Acmequot C Xno ma and f A gt Y is a map and Y H or an interval in R then there exists a continuous function F X gt Y extending 1 Le Fa fa for each a in A Cantor Image Theorem Every separable metric space is the continuous image of a subspace of the Cantor set Baire s Theorem If X is compact Hausdorff or complete metric a countable intersection of dense and open sets is still dense Still to do Mth 6317 Fall 2003 Major Theorems we Mth ear rFall 2003 Major Theorems 26 A First Metrization Theorem Urysohn Metrization Theorem Every regular space X with a countable basis is metrizable Idea of Proof Mimic the proof of the embedding theorem to place X as a subspace of 03 Preview of topics in Mth 632 o Topologies on spaces of functions 46 o Metrization Theorems 3942 0 More on Compactifications 38 o More on Completeness 43 o Fundamental Group and Covering Spaces Ch9 1113 0 Classification of Surfaces Ch 12 Mth 6317 Fall 2003 Major Theorems Mth ear rFall 2003 Major Theorems Homotopy Application of Urysohn s Lemma Def Maps 1 and g from X to Y are homotopic if there Borsuk s Homotopy Extension Theorem exists a map H X X 01 gt Y such that f X gt Uope C H Acmequot C X X X l normal G Agtlt 01 gt Uwith Ga0 fa EIF Xgtlt 01 gt U so that FX0 fX and Fa t Ga t for each a e A HX0 fX and HX 1 gX The map H is called a homotopy from fto g Theorem Homotopy is an equivalence relation on maps from X to Y Example From Homework X is contractible if the identiy map on X is homotopic to a constant map Mm est 7 Fall 2003 Major Theorems 56 Mm est rFall 2003 Major Theorems ee Closed Sets Read Section 17 Closed sets and Limit points Def A subset C of a space X is closed in X if X C if open in X Theorem In a space X 0 The empty set and X are closed 9 Arbitrary intersections of closed sets are closed 0 Finite unions of closed sets are closed Theorem A set C is closed in a subspace Y of X if and only if C is the intersection of Y with a closed subset of X Corollary A closed subset C of a closed subspace Y of X is closed in X Def The closure of a subset A of a space X A or cl XA is the intersection of all closed sets in X that contain A Mth est 7 Fall 2003 Closed Sets Separatlon 18 with est rFall 2003 Closed Sets Separatlon 23 Interior Def The interior of a subset A of X A or lntXA is the union of all open sets in X that are contained in A Note intA C A C clA The first set is open the last closed Theorem If A C Y C X clyA chA n Y Theorem X e A if and only if each basic open set containing X intersects A Terminology neighborhood of X Limit Points Def X is a limit point of A if every neighborhood of X intersects A in a point distinct from X Theorem Let A be the set of all limit points of A Then A AUA Corollary A set C is closed if and only if it contains all of its limit points Mth est 7 Fall 2003 Closed Sets Separatlon Mth est rFall 2003 Closed Sets Separatlon Separation Properties Separation Properties Definitions Def X is if V in X and V in X with o X is a To space given any two distinct points p and q from El Open sets U and Vin X in X either there is an open set containing p and not C U and C V q orthere is an open set containing q and not p o X is a T1 space if given any two distinct points p and q in X there is an open set containing p and not q and 3 there is an open set containing q and not p 4 Note T1 is equivalent to points being closed Def Hausdorff T2 Regular T1 T3 NormalT1 T4 Mtn est 7 Fall 2003 Closed Sets Separation 53 Mtn est rFall 2003 Closed Sets Separation GE Relation to Metric Spaces Properties of Hausdorff Spaces Theorem Metrizable spaces are normal Theorem If X is Hausdorff and p is a limit point of A in X then every neighborhood of p contains infinitely many Def For Y C X dX Y E infdXyy G Y points of A Fads Theorem Linearly ordered spaces are Hausdorff o If Y is closed in X dX Y 0 iff X G Y Subspaces of Hausdorff spaces are Hausdorff The 0 d XX X gt H is continuous product of two Hausdorff spaces is Hausdorff o For fixed Y C X f X gt H given by fX dX Y is Exampleet continuous Mtn est 7 Fall 2003 Closed Sets Separation 73 Mtn est rFall 2003 Closed Sets Separation 33 Read 15 Product Topology and 16 Subspace Topology Finish homework Product Topology Def If X and Y are top spaces the product topology on X X Y is the topology generated by the basis BUgtlt VUopeninXand Vopen in Y Check B is basis for some topology Note Open sets are not only of the form U gtlt V with est 7 Fall 2003 Products Subspaces te Mth est rFall 2003 Products Subspaces 26 Basis for Product Topology Theorem If C is a basis for the topology on X and D is a basis for the topology on Y then EUgtlt VU Cand VG D is a basis for the product topology on X X Y Preview Examples Notation pXXy X PYXrY y Preview 1 Z gt X X Y is continuous if and only if pXo f Z gt X and pyo f Z gt Yare continuous Examples with est 7 Fall 2003 Products Subspaces Mth est rFall 2003 Products Subspaces Subspace Topology Results on Subspace Topology Def If X is a topological space and Y C X the subspace Lemma If B is a basis for the topology for X then topology on Y is the topology consisting of C Um Y U G B is a basis for the subspace topology on Y Um YU is open in X Lemma If U is open in Y and Y is open in X then U is open in X Check This is a topology Theorem If A is a subspace of X and B is a subspace of Examples Y then the product topology on A X B is the same as the subspace topology on A X B as a subspace of X X Y with the product topology Mth est 7 Fall 2003 Products Subspaces 56 with est rFall 2003 Products Subspaces ee Section 22 Quotient Topology Quotient Topology Def A surjective map p X Y is a quotient map if U is open in Y iff p 1 U is open in X Def Given a surjective map p X Y a subset C of X is saturated if p 1pC C Note Being a quotient map is equivalent to being continuous and taking each saturated open set to an open set Note Open and closed maps are quotient maps Examples Def Given p X Space Ase the quotient topology on A is the topology that makes p a quotient map Def Let X be a partition of X into pairwise disjoint subsets Let X be the element of the partition containing X Let p X X be given by pX X If X is given the quotient topology induced by p X is called a quotient space of X Notation XG if G is the partition Exampl g with est 7 Fall 2003 14 Mth est rFall 2003 24 Induced Maps Theorem Induced maps on Quotient Spaces Let p X Y be a quotient map Let Z be a space and g X gt Z be continuous and constant on each p 1 Then g induces a continuous map f Y gt Z such that fo p g Recognizing Quotient Spaces Theorem Recognition of Quotient Spaces Letp X Yand g X gt Z be as in the previous theoremThe induced map f is a homeomorphism iff it is 11 and g is a surjective quotient map X D llZ N with est 7 Fall 2003 34 Mth est rFall 2003 44 Mth 631 Syllabus Fall 2008 Mth 631 Syllabus Fall 2008 Instructor Dennis Garity Contact Information Office Kidder Hall 368 Telephone 7375138 Email garitymathoregonstateedu Office Hours MF 1100 W 100 Others by appointment MWF Web Page wwwmathoregonstateedugarity Text Topology 2nd ed by J Munkres Material Most of chapters 2 4 and parts of 57 Grading HW 150 pts Midterm 100 pts Final 100 pts Test dates Midterm 5th or 6th week Final Two hours date TBA Homework Daily reading assignments and about 5 homework sets Getting Started Chapter 1 review and reference Read Sections 12 and 13 for next time You should not expect to follow everything as it is presented Go over and rewrite your notes after class and go through additional details Mm 6317 Fall zoos memer mo Mm 631 rFall zoos Overview 200 12 I In H dXy E IIX yll 200 yi2 i1 Note For all Xyz in H 0 dX7y 2 0 dX7 0 iff x y 9 dXy dyyx 6 1092 dX7ydyyz We will see these conditions in a more general setting Condition 3 follows from the CauchySchwarz Inequality from Linear Algebra lX39YI S X IIYII Metric Spaces Def A metric space X d is a set X together with a distance function I X X X gt B so that 1 2 and3 from above hold for all Xyz in X Examples Def BgX Def Open set in a metric space X d Exercise Bgx is open in X d Mm 6317 Fall zoos Overview 310 Mm 631 rFall zoos Overview 410 Continuity Limit Points Note Many important concepts in metric spaces such as convergence continuity and limit point can be defined in terms of open sets Convergence Metric definition A sequence Xn in X7 d converges to p if for each a gt 0 there is an N gt 0 so that if ngt N then dX77p lt a Open Set definition Continuity Metric definition A function f Xyd gt Yd is continuous at p if for each a gt 0 there is a 5 gt 0 so that if dX7p lt 5 then dfx7 fp lt a Open Set definition Limit point Metric definition A point p is a limit point of a subset A of X7 d if for each a gt 0 there is a point as E A distinct from p so that d337p lt a Open Set definition Mth est 7 Fall zoos Overview 5l0 Mth est rFall zoos Overview 610 Properties of Open Sets in Xd o X and the empty set are open in Xyd 9 Any union of open sets in X7 d is open in Xyd 0 Any finite intersection of open sets in Xyd is open in X 7d Proof Goals We want to be able to make sense of and prove statements like 0 A sequence in a product HX converges if and only if the components of the sequence converge o A continuous function defined on a closed subset A of B into a space Y can be extended to a continuous function defined on all of H o Compositions of continuous functions are continuous o A space is metrizable if and only if Mth est 7 Fall zoos Overview 7l0 Mth est rFall zoos Overview 510 Topology Questions Distinguish spaces based on intrinsic properties such as o Connectedness o Compactness o Metrizability o Homeomorphism Type 0 Simple Connectivity Topology Questions Determine different ways of placing a space X in a space Y 5 KZQQ MthB l ihiimua OWN sun MthB t iaiimua 0mm tuna Read Sec 20 21 Metric Topology So Far 0 Metric Spaces Open Sets in Metric Spaces 0 Topological Spaces Basis for a Top Space 0 Order Topology 0 Finite Product Topology Subspace Topology 0 Closed Sets and limit Points 0 Continuous Functions 0 General Product Spaces Review definitions of metric space open balls open sets in metric spaces topology generated by a metric A space X is metrizabe if there is a metric d on X generating the topology on X Two metrics d and d on X are equivalent if they induce the same topology A subset A of X d is boundedif El M such that Va and Vb in A da b g M iff El N and a point p in X such that A C BNp Mm e31 rFaiiZOOE Metric Topology vs Mm eai rFaHZOOE MelncTopoiogyri 23 Standard Bounded Metric Defs The diameter of a bounded nonempty subset A of X is supdabab e A The standard bounded metric d corresponding to a metric d on X is definedby d va mindX7y71 Check that d is a metric Theorem Let X d be a metric space and let d be the andard bounded metric corresponding to d Then d and d are equivalent 1 0XHamrceno mnP 0 X R dooOny magtltXi yi I o X equivalence classes of functions from H gt R such that fp du is finite f g if f g ae R eaoltAV QWW VP 0 Any normed linear space X dXy X y Mm est rFaHZOOE Memo Topologyrl Mm eai rFaHZOOE MelncTopoiogyri Equivalent Metrics on R Product Topology Uniform Metric Theorem On H d1d2 and do are equivalent Theorem The product topology on H is the same as the topology induced by d1 d2 and doc Lemma The topology induced by d on X is finer than the topology induced by d if for each X in X and for each 8 gt 0 Proof Show the product topology is the same as the there exists a 5 gt 0 such that topology induced by doc 85X7 d C BgX7 d Def R03 1H The uniform metric on H03 is defined by Proof of Theorem dooX7y S d2X7y S doooriy 3930 SUp d X y where E is the standard bounded metric on H d X7y S d1X7y S nd X7y Mm est rFallZOOE Metric TopologyVi 53 Mm ear rFallZOOE MetricTopologyrl es Comparison of Topologies on R Product Metric The product metric on H03 is defined by Theorem The metric D on H03 induces the product d X7 y topology I DXiy sun I I I I Proof Given BgX7 D choose N so that lt 8 Theorem The topology Induced by the uniform metric on N H the uniform topology is strictly finer than the product topology and strictly coarser than the box topology Given XI 6 W 1 UI X E H i1 iN1 Proof choose 8 lt 1 such that BgX C U Mm est rFallZOOE Metric TopologyVi 73 Mm ear rFallZOOE MetricTopologyrl ss Section 26 Compactness Compactness sections 26 27 28 29 and 45 Def A cover of a space X is a collection of subsets of X whose union is X An open coveris a cover consisting of open sets A subcover of a cover of X is a subcollection of the cover that is still a cover A space X is compact if every open cover of X has a finite subcover Examples Compactness for Subspaces Lemma Let Y be a subspace of X Then Y is compact iff every covering of Y by sets open in X has a finite subcollection covering Y Theorem lf Acmequot C Yeompac then A is compact Theorem Every compact subset C of a Hausdorff space X is closed in X Mm eat 7 Fall 2003 Compactness 13 Mm eat rFall zoos Compactness 23 Relation to Continuity Theorem The continuous image of a compact space is compact Theorem Let f X gt Y be a continuous surjective map where X is compact and Y is Hausdorff Then ftakes closed sets to closed sets If f is 1 1 it is a homeomorphism Other types of compactness X is sequentially compact if every sequence in X has a convergent subsequence X is limit point compact if every infinite set has a limit point X is countany compact if every countable open cover has a finite subcover X is Lindelof if every open cover has a countable subcover Mm eat 7 Fall 2003 Compactness Mm eat rFall zoos Compactness Other Definitions Finite Intersection Property A collection of sets has this property if any finite intersection of the sets is nonempty Complete Every Cauchy sequence converges Totally Bounded For each 8 gt 0 there is a finite covering by 8 balls Lebesgue Number 8 gt 0 is a Lebesgue number of a cover if any set of diameter lt E is contained in some element of the cover Tube Lemma and Finite Products The Tube Lemma Let Y be compact and X be any space If N is an open set containing X0 X Y in X X Y then there is a neighborhood W of X0 in X such that the tube W X Y is contained in N Theorem The product of finitely many compact spaces is compact with est 7 Fall 2003 Compactness 53 with ear rFall zoos Compactness es Countable Products Nested Sequence Lemma If C1 3 Cg D 03 is a nested sequence of closed nonempty sets in a compact space X then 101 0 1 Theorem The countable product of compact metric spaces is compact and metrizable Tychonoff s Theorem Theorem The countable product of compact spaces is compact Tychonoff s Theorem The arbitrary product of compact spaces is compact with est 7 Fall 2003 Compactness Mth ear rFall zoos Compactness Countability Properties Read Sec 20 21 Metric Topology Def A sequence Xn in X converges to p if for each neighborhood U of p there exists an N so that if n gt N then Xn e U Theorem If X is Hausdorff Xn gt p and Xn gt q then P 7 Theorem A countable product of metric spaces with the product topology is metrizable Def A neighborhood base at a point X in X is a collection of neighborhoods of X Um such that every neighborhood of X contains one of the Um Def X is 1st countable if every point in X has a countable neighborhood base X is 2nd countable if X has a countable basis Note Metric spaces are first countable with est 7 Fall 2003 Us with est rFall 2003 26 Sequence Lemma If a sequence of points an in a subset A of X converges to p in X then p e A Conversely if X is first countable and p e A then there is a sequence of points in A converging to p Theorem An uncountable product of nontrivial metric spaces X0 with the product topology is not metrizable Theorem 8 9 is not metrizable Proof The sequence lemma fails The Sequence Lemma Other Examples Theorem R03 with the box topology is not metrizable Proof The sequence lemma fails Theorem Let f X gt Y be continuous where X is 1st countable Then 1 is continuous if and only if whenever an gt a in X fan gt fa in Y Proof True for any X lt Show that rm c with est 7 Fall 2003 36 Mth est rFall 2003 46 Continuity Uniform Continuity Theorem The sum difference product and quotient operations from H X H to H are continuous where defined The sum product difference and quotient of continuous real valued functions are continuous where defined Proof Advanced Calculus Uniform Limit Theorem If Y is metrizable and fn X gt Y is a sequence of continuous functions converging uniformly to a function f X gt Y then fis continuous Another way of stating this is that the subspace of YX consisting of continuous functions is closed in YX if YX is given the uniform bounded metric df g supdfx7 gX Mm est 7 Fall 2003 56 Mm ear rFall 2003 66 Section 27 Compact subsets of R Theorem lf Xlinearly Ordered has the ub property then any closed interval in X is compact Proof Read this proof in the text It is similar to the proof that intervals in a linearly ordered space with the ub property and the betweenness property are connected Corollary Closed intervals in H are compact Note This also follows from the fact that such intervals are complete and totally bounded Compactness in R Theorem A subset A of R is compact if an only if it is closed and bounded with respect to the standard metric on H Max Min Value Theorem Given a map ffrom a compact space X to a linearly ordered space Y there are points 0 and din Xwith fc g fX g fd for all X in X Theorem Let X be nonempty compact Hausdorff If every point of X is a limit point of X then X is uncountable Mm e3i 7 Fall 2003 Compactness in Hquot i3 Mm e3i 7 Fall 2003 Compactness in Rquot 23 We have covered most of the material in Section 28 Theorem Let f be a map from the compact metric space X7 dx to the metric space Y7 dy Then 1 is uniformly continuous Direct construction of Lebesgue number for open covers of compact metric spaces Mm e3i 7 Fall 2003 Compactness in Rquot 33 Section 28 Compactness Implications 1 X is compact if and only if every collection of closed subsets of X with the finite intersection property has nonempty intersection 2 If X is compact then X is countany compact 3 If X is countably compact and Lindelof then X is compact Implications II 4 If X is countably compact then X is limit point compact 5 If X is limit point compact and T1 then X is countably compact 6 If X is limit point compact T1 and first countable X is sequentially compact 7 If X is sequentially compact X is limit point compact Mtn 6317 Fall zoos Compactnoss implications l4 Mtn eat rFaIIZOOS Compactnoss implications 24 Implications III 8 A metric space X d is sequentially compact if and only if it is complete and totally bounded 10 If a metric space X d is sequentially compact then every open cover has an associated Lebesgue number Implications IV 9 If a metric spaces X d is sequentially compact then it is compact Theorem For a metric space X I the following are equivalent compactness countable compactness limit point compactness sequential compactness and the property of being complete and totally bounded Mtn esl 7 Fall zoos Compactness implications Mtn eat rFaIIZOOS Compactnoss implications Sec on19 FWoductTopdogy Def Let Xax J be an indexed family of spaces A point in H Xa will be denoted by Xaa J J 066 Projections onto the coordinate spaces wnaak oteJ are given by BXaa J X3 Note Products can also be viewed as the set of functions from J into UXa such that for each or f0c e Xa BoxTopdogy Exall tpie H Xa where each Xa H also written as H H 0662 Def The box topology on H Xa is the topology with basis OCEJ l H UaUa is open in Xa oteJ lvltn 6317 Fall 2003 General Products le lvlth est rFall zoos General Products 26 ProductTopdogy The product topology on H Xa is the topology with basis OCEJ H Uaan is open in X0 and all but finitely many U0c Xa oceJ Notation Uoc1 gtlt gtlt Uan x H Xa where ot ZJ J a17quot39 7an Bags8ubspaces Theorem lf 806 is a basis for Xa the box and product topologies can be defined using Uoc 6 Ba instead of Ua open in Xa Theorem Products of Subspaces lf Am C Xa then the topology on H Ad is the same as the subspace topology oteJ on H Aor C H Xa if both products are given the box oteJ OCEJ topology or if both products are given the product topology lvltn 6317 Fall 2003 General Products lvltn est rFall 2003 General Products Coninuny of Component Functions Theorem If each Xa is Hausdorff so is H X0c in both the f q E q given by oceJ box and product topologies I 1 ftt7t7t7 39 Theorem Let f A gt H Xa be given by oceJ If the roduct Is Iven the box to olo ma mome P 9 P 9 If the product is given the product topology 1 is continuous if and only if each fa is Mm est 7 Fall 2003 General Prague s 56 Mm est rFall 2003 General premiers ee Baire Spaces An Equivalent Condition Read Section 48 Note A has empty interior if and only each point of A is a limit point of XA if and only if XA X Def A set U is dense in X if U X Lemma X is a Baire space iff given any countable A set is quotOMhire dense in if contains no Open sets collection of dense open sets Un the intersection of or equtvalently A has empty Interior these sets is also dense A Space X is a Baire space if Example Q is not a Baire space R is given any countable collection of nowhere dense closed sets An in X the union of these sets has empty interior Mm 631 7 Fall 2005 Baire Spaces 14 Mm eat 7 Fall 2005 Baire Spaces 24 Baire Category Theorem Applications If X is compact Hausdorff or complete metric then X is 0 El a continous nowhere differentiable function Baire optional section 49 read in text 0 Suppose fn X gt Y d where X is a Baire space Idea of proof Given a collection Fn of closed nowhere If this sequence converges pointwise to f X gt Y depse sets and glven any Open set U Show U contams a then the set of points at which 1 is continous is dense potnt of X UFn in X we will come back to this after talking about pointwise convergence Theorem The set of embeddings from 01 gt Fl is dense in C0 1 Fl with the uniform metric n 2 3 Mm 631 7 Fall 2005 Baire Spaces 34 Mm eat 7 Fall 2005 Baire Spaces 44 Review Material in Chapter 1 Read Sections 12 and 13 Continue Work on Homework Topological Spaces Def A topological space is a set X together with a collection 4 of subsets of X called the open sets so that 0 X and the empty set are open 9 Any union of open sets in is open 9 Any finite intersection of open sets is open Examples 0 W7 d2 where d2 is the usual Euclidean metric 0 Any metric space X7 d with 4 the collection of open sets in Xd Mm esi rFaii zoos Topological Spaces and sases mo Mm esi rFaii zoos Topological Spaces and sases zio More Examples 7 o Rn7 d1 where d1ab Z a b 39 1 o Rn7 doc where dxa7 b maxa b 0 Any set X T X70 or quotrd all subsets of X o H with open sets arbitrary unions of interval of the form a7 b a lt b Notation Hg 0 Finite complement Topology Note Rn7 d2 and W7 d1 have the same open sets so same topology Comparison of Topologies Def A topology 4 on X is finerthan a topology 43 on X if 4 contains 45 Le X743 has more open sets Number of topologies on a7 b7 0 Number of subsets Number of collections of subsets Which are topologies Mm esi rFaii zoos Topological Spaces and sases sio Mm esi rFaii zoos Topological Spaces and sases Aio Basis for a Topology Basis Continued Def Given X a collection C8 of subsets of X is a basis for Theorem Let 4 be the collection of all unions of elements some topology 4 on X if of CB Then 4 is a topology on X 0 If U and V are in CB and if p e Um V there is an set Check WinCBwithpe WCU V 9 Every point in X is in some element of 53 Note The topology 4 is said to be generated by EB Note In the above definition you start with the set X Eggawup rQS being given and determine if B is a basis for some topology on X Mm cal rFall zoos Topologlcal Spaces and sases 5l0 Mm cal rFall zoos Topological Spaces and sases elo Basis foraoivon topology Theorem Let X have a specific given topology 4 Let CB Different questions be a collection of subsets of X so that 0 Given 4 is B a basis for T 0 each set in B is in 4 0 Given B is B a basis for some 4 9 each element of 4 is a union of sets in B General Principal To check whether some property holds Then CB 395 a baSIS and I IS the tOpOIOQy generated by CE for a topology check that it holds for a basis for that Note Condition 2 above is equivalent to tOPOIOQY For each U E 4 and for each p e U there is a B 6 8 with pEBC U Mm cal rFall zoos Topologlcal Spaces and sases 7lo Mm cal rFall zoos Topological Spaces and sases slo Comparison of Topologies Theorem Let CB be a basis for 4 and B be a basis for 45 Then TFAE 1 43 is finer than 4 2 VX in X and VB 6 CB containing X EIB e B with X e B C B Corollary The lower limit topology is strictly finer than the standard topology on B 0 Let 4 be the standard topology for X7 I Let B be the collection of open balls of any positive radius in X 9 Let 4 be the standard topology for X7 I Let B be the collection of open balls of any positive rational radius in X 9 In B5 let B aba lt b Mm est Jan zoos Topological Spaces and Bases 9H0 Mm est 4am zoos Topological Spaces and Bases iono Section 18 Continuity Relation to Metric Definition Def 1 X gt Y is continuous at X if Vbasic open set U in Theorem If the topologies on X and Y arise from metrics Y containing fX El an open set V in X with X e V and this definition of continuity agrees with the metric definition fV C U of continuity The function f is continuous if it is continuous at each point Assume f X gt Y is continuous at X according to the topological definition This is equivalent to the following condition Conversely V basic open set U in Y f 1 U is open in X Mtn est 7 Fall 2003 Continuity i3 Mtn est rFall 2003 Continuity 23 Equivalent Formulations of Continuity Homeomorphisms and Embeddings Theorem Given 1 X gt Y TFAE Def A bijection f X gt Y is called a homeomorphism if 0 fig continuous Lei both land f 1 are continuous f 1 U is open in X whenever U is open in Y e y subset A of X rA c fA Homeomorphism is the basic notion of equivalence in topology 0 V closed set B in Y f 1 B is closed in X 12 Esalnples X G A eaCh nbhd V 0f 0 39nterseCts f Def A topological embedding of X in Y is a 23 homeomorphism f X gt fX C Y If A r1 B show A c A 31 Mtn est 7 Fall 2003 Continuity 33 Mtn est rFall 2003 Continuity 43 Knots as Embeddings of 81 Constructing continuous functions Constant functions are continuous Inclusions of subspaces are continuous Compositions preserve continuity Restricting the domain or expanding the range of continuous function gives a continuous function a f X gt Y is continuous if X Ungen and for each 06 0000 x f U0c is continuous f Xe Y is continuous if for each X in Xand for each open neighborhood V of fX there is a neighborhood U of X with fU C V 0 M m hi mm mm ya M an m mm mm ma Pasting Lemma Maps into Products Continuity of Distance Functions Pasting Lemma If X AU B where A and B are closed Theorem If the topology on X arises from a metric of then in X F A a Yand g B a Y are continuous with f g onA B then 0 dXgtltXgtF1 iS continuous h 7 fX if X E A 9 For fixed Y f Xe B given by fX d7 Y is 7 goo X E B continuous 1 Let ab 6 Xgtlt Xand Bsfab be given is continuous 2 Maps into products f A a X gtlt Y given by fa f1a7f2a is continuous iff f1 and f2 are continuous Mth t ihiimua 00mm 7m MthB t iaiimua 00mm 18 Section 24 Connectedness in Ft Fi is Connected Def A linearly ordered set L having more than one point is Cor H is connected as is every interval and ray in H a linear continuum if u I intermediate Value Theorem If 1 Is a map from a L has the IUb Property connected space S into an ordered space Y and if 0 If X lt 2 there is a y with X lt y lt Z fa lt r lt fb then there is a point c in X with fc r I I I I Def Apath in X from X to y is a map f a7 b gt X such Theorem A linear39contlnuu39m LWIth the order topology Is that ma X and Kb y X is path connected for each ConneCted and so 395 every 39nterval and ray m L39 pair of points X and y in X there is a path from X to y Examples Mm est 7 Fall 2003 Connectedness in R 13 Mm est rFall zoos Connectedness in R Path Connected vs Connected Theorem X path connected X connected X connected qtgt X path connected Examples Mm est 7 Fall 2003 Connectedness in R Compact 1 9 2 gt 4 3 Lindelof Compactness Definitions and Implications 4 Countany gt Limit Pt Compact 4 Compact 5Ti 9metric Compact Every open cover has a nite subcover 6 T1lstctb1 gt 7 metric Sequentially H Compact 8 10 metric Every open cover has a Lebesgue number Every collection of closed sets With the finite intersection property has nonempty intersection Finite Intersection Property A collection of sets has this property if any nite intersection of the sets is nonempty Lindelof Every open cover has a countable subcover Countably Compact Every countable open cover has a nite subcover Limit Point Compact Every in nite set has a limit point Sequentially compact Every sequence has a convergent subsequence Complete Every Cauchy sequence converges Totally Bounded For each sgt0 there is a nite covering by 8 balls Lebesgue Number sgt0 is a Lebesgue number of a cover if any set of diameter lt s is contained in some element of the cover Complete amp Totally de Section 23 Connectedness Def A separation of a space X is a pair U V of disjoint nonempty open sets whose union is X X is disconnected if there exists a separation of X Otherwise it is connected Note X is connected iff the only subsets that are both open and closed are X and Q Examplee Connected Subspaces Lemma If Y is a subspace of X Y is connected iff nonempty subsets A and B of Y whose union is X neither of which contains a limit point of the other Examples Lemma If C and D form a separation of X then any connected subspace of X lies entirely in C or in D Mm est 7 Fall 2003 connectedness 14 Mm est 7Fall zoos Connectedness 24 Relation to Closure Theorem The union of a collection of connected subspaces that have a point in common is connected Theorem If A C B C A and A is connected so is B Theorem The product of connected spaces is connected Def map E continuous function Mm est 7 Fall 2003 connectedness Mm est rFall zoos connectedness 29 Local Compactness Def X is locally compact at X if there is a compact subset C of X that contains a neighborhood of X Def Let X be locally compact Hausdorff Let Y be XU p for some p X Define U be open in Y if U C X is open in X or if U Y C for some compact C in X Y is called the one point compactification of X Note This defines a topology on Y and is of interest only when X is not compact One Point Compactification Theorem Let X be locally compact Hausdorff Let Y be the one point compactification of X Then Y is compact Hausdorff X is a subspace of Y YX consists of a single point Note If X is not compact X Y Def Compactification Alternate Form of Local Compactness If X is Hausdorff X is locally compact at X iff for each nbhd U of X there is a nbhd V of X with V compact and V in U Mlh e3l 7 Fall 2003 Local Compaclness l3 Mlh e3l rFall 2003 Local Compaclness 23 Corollary Let X be locally compact Hausdorff If Y is closed or open in X then Y is locally compact Theorem X is homeomorphic to an open subset of a compact Hausdorff space iff X is locally compact Hausdorff Mlh e3l 7 Fall 2003 Local Compaclness 33 Characterization De nitions for Mth 631 Midterm Metric Space Bgx Open set in a metric space Topology Topology induced by metric Linearly Ordered Set Order Topology Dictionary Order Topology Subspace Topology Separable Basis for some topology Well ordered Product topology on X gtlt Y R Closed Set Limit Point Closure Interior Boundary Product Toplogy on H Xa Box Toplogy on H Xa are are Equivalent Bounded Metric Uniform Metric on HXada a Product Metric on 110ng i1 SQ Convergent sequence Continuous function First countable Second countable Quotient topology Quotient map Quotient space Dense T0 T1 T2 T3 T4 Hausdorff Regular Normal Basis for given topology Continuous at a point De nitions for Mth 631 Final Metric Space Bgx Open set in a metric space Topology Topology induced by metric Linearly Ordered Set Order Topology Dictionary Order Topology Subspace Topology Separable Basis for some topology Well ordered Product topology on X gtlt Y R Closed Set Limit Point Closure Interior Boundary Product Toplogy on H Xa Box Toplogy on H Xa are are Equivalent Bounded Metric Uniform Metric on HXada a Product Metric on 110ng i1 SQ Convergent sequence Continuous function First countable Second countable Quotient topology Quotient map Quotient space Dense T0 T1 T2 T3 T4 Hausdorff Regular Normal Basis for given topology Continuous at a point connected separation component path conected path component compact Lindelof countably compact limit point compact sequentially compact complete totally bounded Lebesgue number locally connected locally path connected locally compact compacti cation locally connected disconnected one point compacti cation Urysohn Function

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