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Math 131: Introduction to Topology 1

Math 131: Introduction to Topology1 Professor Denis AurouxFall, 2019 Contents9/4/2019 - Introduction , Metric Spaces, Basic Notions39/9/2019 - Topological Spaces, Bases99/11/2019 - Subspaces, Products, Continuity159/16/2019 - Continuity, Homeomorphisms, Limit Points219/18/2019 - Sequences, Limits, Products269/23/2019 - More Product Topologies, Connectedness329/25/2019 - Connectedness, Path Connectedness379/30/2019 - Compactness4210/2/2019 - Compactness, Uncountability, Metric Spaces4510/7/2019 - Compactness, Limit points , Sequences4910/9/2019 - Compactifications and Local Compactness5310/16/2019 - Countability, Separability, and Normal Spaces5710/21/2019 - Urysohn s Lemma and the Metrization Theorem611 Please email Beckham Myers at with any corrections, questions, or comments.

11/6/2019 - The Brouwer Fixed Point Theorem85 11/11/2019 - Antipodes and the Borsuk-Ulam Theorem88 11/13/2019 - Deformation Retracts and Homotopy Equivalence91 11/18/2019 - Computing the Fundamental Group95 11/20/2019 - Equivalence of Covering Spaces and the Universal Cover99 11/25/2019 - Universal Covering Spaces, Free Groups104

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Transcription of Math 131: Introduction to Topology 1

1 Math 131: Introduction to Topology1 Professor Denis AurouxFall, 2019 Contents9/4/2019 - Introduction , Metric Spaces, Basic Notions39/9/2019 - Topological Spaces, Bases99/11/2019 - Subspaces, Products, Continuity159/16/2019 - Continuity, Homeomorphisms, Limit Points219/18/2019 - Sequences, Limits, Products269/23/2019 - More Product Topologies, Connectedness329/25/2019 - Connectedness, Path Connectedness379/30/2019 - Compactness4210/2/2019 - Compactness, Uncountability, Metric Spaces4510/7/2019 - Compactness, Limit points , Sequences4910/9/2019 - Compactifications and Local Compactness5310/16/2019 - Countability, Separability, and Normal Spaces5710/21/2019 - Urysohn s Lemma and the Metrization Theorem611 Please email Beckham Myers at with any corrections, questions, or comments.

2 Anymistakes or errors are - Category Theory, Paths, Homotopy6410/28/2019 - The Fundamental Group(oid)7010/30/2019 - Covering Spaces, Path Lifting7511/4/2019 - Fundamental Group of the Circle, Quotients and Gluing8011/6/2019 - The Brouwer fixed Point Theorem8511/11/2019 - Antipodes and the Borsuk-Ulam Theorem8811/13/2019 - Deformation Retracts and Homotopy Equivalence9111/18/2019 - Computing the Fundamental Group9511/20/2019 - Equivalence of Covering Spaces and the Universal Cover9911/25/2019 - Universal Covering Spaces, Free Groups10412/2/2019 - Seifert-Van Kampen Theorem, Final Examples10929/4/2019 - Introduction , Metric Spaces, Basic NotionsThe instructor for this course is Professor Denis Auroux. His email is his office is SC539.

3 He will be hosting office hours Monday 12:30-2 and Tuesday 9-10:30. Thecourse website is auroux/131f19/. All information will be posted on thecourse webpage, although we will use Canvas to record will be homework due every week on Wednesday, along with a take-home midterm andan in class final. We will loosely follow Munkres Topology . The only prerequisites are some famil-iarity with the notion of a group and some comfort with metric spaces/the ability to manipulateopen and closed , geometry is the study of measuring quantities. Mathematicians then use these measure-ments to make conclusions about properties of the spaces being studied. Topology , on the otherhand, studies spaces by asking questions from a qualitative perspective.

4 For example, some topo-logical questions include: Is a space connected? Is a spacesimplyconnected? This question provides a technique for distinguishing betweena sphere and a torus. For on the torus, there exist closed curves which cannot be shrunk toa point. Is a spaceoriented? For example, the regular cylinder is oriented (as it has two sides), whilethe M obius space is not (it has only one side). Note that there are easier ways to distinguishthese two, namely by examining their topologyis the field that studies invariants of topological spaces that measure these aboveproperties. For example, thefundamental groupmeasures how far a space is from being simplyconnected. Before this, however, we will develop the language of point set Topology , which extendsthe theory to a much more abstract setting than simply metric we will remain informal, but atopological spaceis an abstraction of metric spaces.

5 Inshort, a topological space is a set equipped with the additional data necessary to make sense ofwhat it means for points to be close to each other. This will allow us to develop notions of limitsand Power of Abstraction - Example from AnalysisWe have the following classical theorem:Theorem(The Extreme Value Theorem).Given a continuous functionf: [a,b] R,fachieves is maximum and minimum in the interval[a,b].This theorem can be generalized to the a continuous functionf:C Rfrom a compact setC,fachieves itsmaximum and minimum this is itself a special case of an even more general a continuous functionf:C Xfrom a compact setCto a topologicalspaceX, the image offis is one excellent example of the power of abstraction, as we can take existing results andexpand them to vastly more generalize will introduce metric spaces in order to motivate the definition of topological spaces (otherwise,the definition seems a bit arbitrary).

6 Metric spaces and open spaceis a pair(X,d), whereXis a set andd:X X R 0is thedistance (x,x) = 0andd(x,y)>0whenx6=y, for allx,y (x,y) =d(y,x), namelydis (x,z) d(x,y) +d(y,z)for allx,y,z X. This is thetriangle inequality, which saysthat the shortest path between two points is the straight line between The vector spaceRnwith the Euclidean distanced(x,y) = n i=0(yi xi)2wherex= (x1,..,xn),y= (y1,..,yn) Rn, is a metric space. This is the usualdistance in space. It s easy to check that this indeed defines a metric on the spaceRn. LetY Rn. ThenYbecomes a metric space under theinduced metric. In particular,we define a metric onYby simply restricting the metricd|Y that this is not always the appropriate metric to use on a subspace.

7 For exam-ple, the surface of the earth is a subset of space, but we don t usually measure thedistance between two points on the earth by simply drawing a straight line between themin space. We can define another metric onRnby takingd (x,y) = maxi|yi xi|You will check this a metric on the first homework. We can define another metric onRnby takingd1(x,y) =n i=1|yi xi|Once we have a notion of distance, we can discuss open sets. The idea of topological spaces will beto bypass the notion of distance and simply consider these open a metric space(X,d)and a pointp X, theopen ballof radiusr R>0aroundpisBr(p) ={q X:d(p,q)< r}Such an open ball is sometimes referred to as theopen neighborhoodofpof balls are instances ofopen subsetU Xisopenif, for every pointx U, there exists >0such thatB (x) idea is that, in a open set, there exists a safety margin around every point.

8 Given a pointp,one can move around in the set a certain distance and remain in the basic properties of open sets are1. Open balls are open. This is a basic consequence of the triangle inequality. It is on the is open (vacuously). open (as all open balls are contained inX).4. The arbitrary union of open sets is open (even infinitely many). This follows easily from The intersection of finitely many open sets is is important to note that we can only expect that the intersection of finitely many open sets isstill open. For example, open intervals are open inR, but the intersection n N( 1n,1n)={0}is not open (as there are no open balls around 0 contained in{0}).Limits and closed setsOne very important notion in the theory of metric spaces is that of asequence.

9 Let (X,d) be ametric sequencep1,p2,.. Xconvergesto alimitp Xif, for all >0, there existssomeN Nsuch that for alln Nwe haved(pn,p)< . a sequence in a metric space converges to a limit, this limit is is false in a general topological space. We will discuss the properties of a topological spacethat will guarantee a sequence has a unique can formulate the notion of the convergence of a sequence without mentioning the limit this case, we want that the points of a sequence become arbitrarily close to each other (whereasabove, we demanded that the points become arbitrarily close to a given pointp). sequencep1,p2,.. XisCauchyif, for all >0, there existsN Nsuch thatfor alln,m Nwe haved(pn,pm)< .It is easy to prove that a converging sequence is Cauchy using the triangle inequality.

10 The idea isthat, if all the points are becoming arbitrarily close to a given pointp, then they are also becomingclose to each other. The converse is not always true, metric space iscompleteif every Cauchy sequence also converges to a fact, every metric spaceXis sitting inside a larger, complete metric a metric spaceX, one can construct thecompletionof a metric space by consid-ering the space of all Cauchy sequences inXup to an appropriate equivalence relation. Then thisspace of Cauchy sequences is itself a metric space which restricts to the original metric setZ Xisclosedif the complementX\Zis subset does not need to be open or closed. Subsets can be open, closed, open and closed,or neither open nor example, andXare always both open and closed.


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