Topology - people.math.harvard.edu

1 TopologyCourse Notes Harvard University Math 131 Fall 2013C. McMullenContents1 Introduction ..12 Background in set theory ..33 Topology .. 124 Connected spaces .. 235 Compact spaces .. 276 Metric spaces .. 327 Normal spaces .. 418 Algebraic Topology and homotopy theory .. 459 Categories and paths .. 4710 Path lifting and covering spaces .. 5311 Global Topology : applications .. 5812 Quotients, gluing and simplicial complexes .. 6113 Galois theory of covering spaces .. 6614 Free groups and graphs .. 7515 Group presentations, amalgamation and gluing .. 811 IntroductionTopology is simply geometry rendered flexible. In geometry and analysis, wehave the notion of a metric space, with distances specified between if we wish, for example, to classify surfaces or knots, we want to thinkof the objects as a topologist, all triangles are the same, and they are all thesame as a circle.

2 For a two dimensional example, picture a torus with a hole1in it as a surface inR3. This space can be almost completely flattened , it is topologically the same as a rotary with an underpass connectingits inner and outer 1. Rotary with we take an ordinary band of paper and put in a twist, it becomes aM obius band, which seems clearly different, but how so? What happens ifyou cut a M obius band down the middle? What happens if you do it again?Is a configuration of two linked circles in space fundamentally different fromtwo unlinked circles?The idea of a topological property we want to maintain ina topological space is that ofnearness. We will allow shapes to be changed,but without tearing them. This will be codified by open underlies all of analysis, and especially certain large spaces suchas the dual ofL (Z) lead to topologies that cannot be described by spaces form the broadest regime in which the notion of acontinuous functionmakes sense.

3 We can then formulate classical and basictheorems about continuous functions in a much broader example, an important theorem in optimization is that any continuousfunctionf: [a,b] Rachieves its minimum at least one pointx [a,b].This property turns out to depend only oncompactnessof the interval, andnot, for example, on the fact that the interval is finite second agenda in Topology is the development of tools to telltopological spaces apart. How is the M obius band to be distinguished fromthe cylinder, or the trefoil not from the figure eight knot, or indeed how isR3different fromR4? Our introduction to the tools ofalgebraic topologyprovides one approach to answer these course correspondingly has two parts. Part I ispoint set Topology , which is concerned with the more analytical and aspects of thetheory.

4 Part II is an introduction toalgebraic Topology , which associatesalgebraic structures such as groups to topological will follow Munkres for the whole course, with some occassional addedtopics or different will consider topological spacesaxiomatically. That is, a topologicalspace will be a setXwith some additional structure. Because of the gener-ality of this theory, it is useful to start out with a discussion of set on writing you hit a home run, you just have tostep once on the center of each base as you round the field. You don t haveto circle first base and raise a cloud of dust so the umpire can t quite see ifyou touched the base but will probably give you the benefit of the Background in set theoryThe axioms of set theory. Axiom I. (Extension) A set is determined by its elements.

5 That is, ifx A= x Band vice-versa, thenA=B. Axiom II. (Specification) IfAis a set then{x A:P(x)}is also aset. Axiom III. (Pairs) IfAandBare sets then so is{A,B}. From thisaxiom and = 0, we can now form{0,0}={0}, which we call 1; andwe can form{0,1}, which we call 2; but we cannot yet form{0,1,2}. Axiom IV. (Unions) IfAis a set, then A={x: B,B A&x B}is also a set. From this axiom and that of pairs we can form {A,B}=A B. Thus we can definex+=x+1 =x {x}, and form, for example,7 ={0,1,2,3,4,5,6}. Axiom V. (Powers) IfAis a set, thenP(A) ={B:B A}is also Axiom VI. (Infinity) There exists a setAsuch that 0 Aandx+ 1 Awheneverx A. The smallest such set is unique, and we call itN={0,1,2,3,..}. Axiom VII (The Axiom of Choice): For any setAthere is a functionc:P(A) { } A, such thatc(B) Bfor allB of the I.

6 (Extension).To have a well defined domain of discourse, the elements of sets arealso is a subtle point in Axiom I: what does the conclusion,A=B,mean anyway? In fact the idea of equality is a notion in logic rather thanset theory. It means that for any logical sentenceP(x),P(A) has the sameanswer asP(B). For example, ifA=B, andA Y, thenB II. (Specification).Examples:A B={x A:x B}.A B={x A:x6 B}.R Q= irrationals;Q R= .{x Z: y Z,y+y=x}= even numbers.{x Z:x/n Z n >0}={0}.{x Z:x2<0}= .For more advanced set theory, one uses the Axiom of Replacement insteadof Specification; this permits the construction of cardinals such as , but itis not required for most mainstream at least one setAexists, we can now form0 = ={x A:x6=x},but nothing else for sure. ( be .)The Barber of Seville; Russell s {A:A6 A}, isX X?

7 There is no universe: given a setA, setX={B A:B6 B}.We claimX6 A. Indeed, ifX A, thenX XiffX6 solution to the classic paradox who shaves the barber of Seville? is of course that the barber is a woman. In the G odel-Bernays theory, youare allowed to formX, butXis not a set; it is called a III. (Pairs).From this axiom and = 0, we can now form{0,0}={0}, which we call 1; and we can form{0,1}, which we call 2; but we cannotyet form{0,1,2}.4 Axiom IV. (Unions).From this axiom and that of pairs we can form {A,B}=A B. Thus we can definex+=x+ 1 =x {x}, and form, forexample, 7 ={0,1,2,3,4,5,6}. , we can define A={x: B A,x B}. SinceAhas at least one elementB0, we have A B0and thus the intersectionis a set. Note: is undefined!Examples: {A}=A, {A,B}=A V. (Powers)Examples:X={B P(52) :Bhas exactly 5 elements}is the number of possible poker hands.

8 |X|= 2,598, s triangle. The subsets withk+ 1 elements of{1,..,n}can bepartitioned into those that includenand those that do not. Thus(n+1k)=(nk)+(nk 1).Axiom VI. (Infinity). We have now built up the natural numbers via settheory, and can proceed to the real numbers, functions on them, etc., witheverything resting on the empty standard assumption we have not listed is the Axiom of Exten-sion, which asserts there is no decreasing x2 x1. Thisimpliesallsets rest on the empty set, and we never have theinfinite loopx kindly mathematician uncle asks his niece, What s the highest numberyou know? The niece replies, 168,000,000 . The uncle asks, But what about168,000,001 ? And the niece replies, I was close, wasn t I? (Hubbard)Moving can now defineordered pairsby(a,b) ={{a},{a,b}}.

9 Then (a,b) = (a ,b ) iffa=a andb=b . Then we can define theproductoftwo sets byA B={(a,b) :a A,b B}.Note thatA B P(P(A B)), so it is a a subsetR A B. It has relation can be visualized as a directed graph with verticesA Bandwith an edge fromatobexactly when (a,b) : an equivalence relation is a subset ofA Awith certainproperties. The relationi < jonZ. The relationb|aon{1,2,..,10}. :A Bis a relation betweenAandBsuch thatfor eacha A, there is a uniquebsuch that (a,b) f. We write this asb=f(a). Functions are also set of allf:A Bis denotedBA. Why? How many elements does35have? (Answer: 243.)A function can beinjectiveand/orsurjective. It isbijectiveif of maps:f g. Iff:A Bis bijective, then there is aunique mapg:B Asuch thatg f(x) =x x :f(n) =n2is injective onN, but not onZ. It is surjective inneither case.

10 The function sin :R [ 1,1] is surjective but not restriction, sin : [ /2, /2] [0,1], is bijective. Its restriction, sin :[0,1] [ 1,1], is injective but not theory as a programming point of the definitions ofNand (a,b) is not so much that they are natural or canonical, but that theywork. In other words set theory provides a very simple language in whichthe rest of mathematics can is a natural bijection betweenA is a natural bijection betweenP(A) and (X) as an we defineA B=A BandA+B= (A B) (A B), thenP(X) becomes a ring. The identity elements are rings is nothing but the ring of mapf:X thatA+A= 0. Thus this ring is also analgebraover the let :P(X) 2 Xdenote the map that sendsAto itsindicator function Awhich is 1 onAand otherwise , unions and :X Ybe a function. Wesetf(A) ={f(a) :a A}.