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Topology Notes Topology Lectures –Integration workshop 2017 David Glickenstein August 2, 2017 Abstract Lecture notes from the Integration Workshop at University of Ari- zona, August 2017. These notes are based heavily on notes from previous integration workshops written by Philip Foth, Tom Kennedy, Shankar Venkataramani and others. 1 Introduction to topology 1.1 Topology theorems The basics of point set topology arise from trying to understand the following theorems from basic calculus: (in the following, we assume intervals written [a; b] have the property a < b so that they are not empty) Theorem 1 (Intermediate Value Theorem) If f :[a; b] R is a contin- uous function, and y is a number between f (a) and f (b) !then there exists x [a; b] such that f (x) = y: 2 Theorem 2 (Extreme Value Theorem) If f :[a; b] R is a continuous ! function, then there are numbers xm; xM [a; b] such that 2 f (xm) = min f (x): x [a; b] ; f 2 g f (xM ) = max f (x): x [a; b] : f 2 g Here are some other interesting theorems in topology that we will not prove here: Theorem 3 (Jordan Curve Theorem) Any continuous simple closed curve in the plane, separates the plane into two disjoint regions, the inside and the outside. Theorem 4 (Jordan-Schoen‡eiss Theorem) For any simple closed curve in the plane, there is a homeomorphism of the plane which takes that curve into the standard circle. 1 Theorem 5 (Generalized Jordan Curve Theorem) Any embedding of the (n 1)-dimensional sphere into n-dimensional Euclidean space separates the Euclidean space into two disjoint regions. Theorem 6 (Brouwer Fixed Point Theorem) Any continuous function from the closed disk in Rn to itself has a …xed point. Theorem 7 (Borsuk-Ulam Theorem) If f : S2 R2 is continuous, then there exists a point x S2 such that f (x) = f ( x) :! 2 Theorem 8 (Invariance of Dimension) No nonempty open subset of Rn is homeomorphic to an open subset of Rm if m = n: 6 1.2 Topology of Rn For motivation, we recall what open and closed sets look like in Rn. A set U Rn is open if for any x U there is a ball centered at x contained in U: A set F is closed if it contains2 all of its limit points, i.e., for every convergent sequence that is in F , the limit is in F . It can be shown that a set in Rn is closed if and only if its complement is open. Let f be a function from Rn to Rk or from U to Rk where U is an open set n in R . Then the de…nition of continuity for f at x0 is that > 0, > 0 8 9 such that x x0 < implies f(x) f(x0) < . (Here denotes the jj jj jj jj jj jj usual distance function in Rn or Rk.) If f is continuous at every point in its domain we say it is continuous. It then follows that f is continuous under this k 1 de…nition if and only if, for all open subsets U in R , f (U) is open in Rn. The only structure of Rn that we need in the above is the ability to measure the distance betweeen two points in the space. So we can immediately generalize the above to a metric space. The above shows more, namely that we can do a lot if we just know what the open sets are, not the metric they came from. So we can abstract things by just looking at the collection of open sets. Note that the open sets in Rn, and more generally the open sets in a metric space, have some obvious properties. The empty set and the whole space are open. Any union of open sets is an open set. Any …nite intersection of open sets is open. These observations will be the basis for the de…nition of a topology. 1.3 Metric spaces De…nition 9 A metric space (X; d) is a set X and a function (called the metric) d : X X R such that for all x; y; z X; the metric satis…es: ! 2 1. (positive de…nite) d (x; y) 0 with d (x; y) = 0 if and only if x = y 2. (symmetric) d (x; y) = d (y; x) 3. (triangle inequality) d (x; z) d (x; y) + d (y; z) 2 De…nition 10 In a metric space (X; d), a set U is said to be open if x U, > 0 such that d(y; x) < implies y U. 8 2 9 2 Proposition 11 Let (X; d) be a metric space and let be the collection of open sets in X. Then T 1. X and ? , 2 T 2 T 2. Arbitrary unions of sets U are in , i.e., for any indexing set I; if 2 T T Ui for all i I then Ui , 2 T 2 2 T i I [2 3. If U; V then U V . 2 T \ 2 T For any set X, we will de…ne a collection of subsets of X to be a topology for X if it satis…es the three properties above. Note thatT property 3 immediately implies by induction that a …nite intersection of open sets produces an open set. De…nition 12 A sequence xn in a metric space (X; d) converges to a point x X if > 0, there exists an index N < such that n > N = d(xn; x) < . 2 8 1 ) De…nition 13 A subset F of a metric space (X; d) is closed if for every se- quence xn in F which converges to some x in X we have x F . 2 Proposition 14 A set F is closed if and only if F C = X F is open. n Corollary 15 Arbitrary intersections and …nite unions of closed sets are closed. De…nition 16 The interior of a set A; denoted A or int(A), is int(A) = x : > 0 s:t: d(x; y) < = y A f 9 ) 2 g The closure of a set A; denoted A or cl(A), is cl(A) = x : > 0 y A s:t: d(x; y) < f 8 9 2 g Proposition 17 The interior of a set A is the union of all open sets contained in A: The closure of a set A is the intersection of all closed sets containing A. Note that the proposition shows that the interior is open since it is a union of open sets, and the closure is closed since it is an intersection of closed sets. If the space X is a vector space, then one way to get a metric on X is to start with a norm. De…nition 18 A function on X is a norm if jj jj For all x X, x 0, and x = 0 if and only if x = 0. 2 jj jj jj jj ax = a x for every a R and x X. jj jj j jjj jj 2 2 3 x + y x + y for every x; y X. jj jj jj jj jj jj 2 A normed space is a vector space with a norm de…ned on it. Proposition 19 Let (X; ) be a normed space. De…ne d(x; y) = x y . Then (X; d) is a metric space.jj jj jj jj We end this section with some examples of metric spaces. 1. Euclidean metric on Rn: The usual Euclidean norm gives a metric on Rn. 1=2 n 2 d(x; y) = x y = xj yj jj jj 2 j j 3 j=1 X 4 5 2. R with a di¤erent topology: De…ne a metric on X by x y d(x; y) = j j 1 + x y j j One of the homework problems is to check this is a metric and to determine if it gives a di¤erent topology for R from the standard one. 3. Space of functions: Let D be any set and let X be the set of all bounded real-valued functions on D. De…ne d(f; g) = sup f(x) g(x) x D j j 2 Then (X; d) is a metric space. 4. lp norm on Rn: There are other norms we can put on Rn and hence other metrics. For 1 p < , de…ne 1 1=p n p x p = xj jj jj 2 j j 3 j=1 X 4 5 (The case p = 2 is the usual Euclidean metric.) It is not hard to show that we get the same collection of open sets, i.e, the same topology, for all the value of p. As p we get ! 1 5. sup norm on Rn: The function d(x; y) = max xj yj 1 j n j j is another metric on Rn that de…nes the same toplogy. 4 6. lp(N): If we look at in…nite sequences instead of just vectors, things are p p more interesting. Let l be the set of sequences (xn)n1=1 with n1=1 xn < . For such a sequence we de…ne j j 1 P 1=p 1 p (xn)n1=1 p = xn jj jj "n=1 j j # X Consider the two sets p F = (xn)n1=1 l : xn 0 n f 2 p 8 g U = (xn)1 l : xn > 0 n f n=1 2 8 g Is F closed? Is U open? 7.
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