TOPOLOGY I M 367K
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Date Created: 09/06/15
Chapter 2 General Topology In this chapter we will start with the real number line and investigate some of its properties We will then de ne a topological space as an abstraction of features of the real line The topological ideas of limit point convergence open and closed sets and continuity are all the result of capturing essential characteristics that we nd in the real numbers 21 The Real Number Line We will not present an axiomatic de nition of the real numbers Instead we will rely on our understanding of the real number line as the set of all decimal numbers ordered in their familiar way Let us rst review the concepts necessary to de ne convergence of se quences and continuity of functions on the real number line De nition open interval In the real number line R de ne an open interval a b as the set x E Rla lt z lt 1 De nition open interval centered at Given m E R and e gt 0 the open interval centered at z of radius 6 Bz e is the open interval m 7 e z 6 De nition open set in R In R a set U is open if and only if for every point z E U there is an em gt 0 such that m 7 ehz em Q U Theorem 21 The empty set is open and R is open Theorem 22 If U1 and U2 are open sets then U1 U2 is open In fact the intersection of nitely many open sets is open Theorem 23 The union of any collection of open sets in R is open Theorem 24 If U is open then U is the union of open intervals 11 12 CHAPTER 2 GENERAL TOPOLOGY Let us recall the de nition of convergence of a sequence in R De nition convergent sequence We say a sequence uhEN Q R converges to m or that z is the limit of the sequence written as x 7 x if and only if for every 6 gt 0 there is an N E N such that lz 7 lt e for all i gt N We are in the process of recasting ideas from the real line in set theoretic terms So let s rephrase the de nition of convergence without using distance but instead in terms of open intervals The de nition above says that any open interval of radius 6 centered at z contains all but nitely many of the elements of the sequence Instead of restricting ourselves to open intervals centered at m we can consider any open set containing z and de ne convergence as follows De nition convergent sequence lf miheN Q R then we say that m con verges to z written as x 7 x if and only if for any open set containing m there is an N E N such that z E U for all i gt N If instead of studying sequences we wish to look at sets in R in general we would like to de ne what we mean by saying that a point is close to other points in that set One way is to abstract the concept of distance but we don t actually need to have a distance we can also use the more general concept of an open set Let us see how it would play out in R We could say that z is close to a set A if there is a sequence of elements of A that converges to m If x is in A however we could always cheat and pick the constant sequence where z z for all or almost all i This de nition would not include the idea that x has nearby points from A To avoid this problem we could say that z is close to a set A if there is a sequence of elements of A 7 that converges to z But a more general de nition would be to dispense with sequences altogether and use the condition that the intersection of any open set containing z with A7 is never en1pty So we thus come to the following de nition De nition limit point in R Let A be a subset of R and x be a point in R Then z is a limit point of A if and only if for every open set U containing x U 7 O A is not empty In other words a limit point of a set is one that cannot be isolated from the rest of the set with an open set De nition closed set in R A set in R is closed if and only if it contains all of its limit points Theorem 25 The intersection of any collection of closed sets in R is closed Theorem 26 The union of two closed sets in R is closed In fact the union of nitely many closed sets in R is closed 22 OPEN SETS AND TOPOLOGIES 13 Closed sets and open sets are related by the following theorem Theorem 27 A set in R is open if and only if its complement is closed Let us now review the de nition of continuity that you probably rst encountered in calculus7the 6 6 de nition De nition continuous function in R A function f D g R 7gt R is con tinuous at x if and only if for every 6 gt 0 there is a 6 gt 0 such that if z E D and lz 7 zl lt 6 then 7 lt e We say f is continuous if it is continuous for every point z in its domain D We wish to convert this de nition into the language of open sets and intervals It is saying that if a function is continuous in its domain then if we pick an open interval I 7 e fz 6 then we can nd an open interval J m 7 6m 6 that is mapped into I So in general if we pick an open set U Q R that contains a point fm then we can nd an open set V Q R containing z whose image is contained inside U This set theoretic view of continuity allows us to reword the concept of continuity in the language of open sets as follows De nition continuous function in R A function f R 7gt R is a continuous function if and only if for every open set U in R f 1U is open in R Theorem 28 The two de nitions of continuity for real valued functions on R are equivalent 22 Open Sets and Topologies We have now seen that several important concepts in analysis convergence limit points closed sets and continuity can be de ned using ideas about sets and their intersections and unions In our familiar world of the real numbers open sets were the central players in all these concepts Now we would like to extend these concepts to spaces other than the familiar real numbers with their usual concept of open set Our strategy is to abstract a more general concept of an open set from our experience with the real numbers To that end we isolate some of the conditions that were satis ed by the usual open sets of R and use those properties to de ne a topology and a topological space De nition topology Suppose X is a set Then T is a topology for X if and only if T is a collection of subsets of X such that 1 WET 14 CHAPTER 2 GENERAL TOPOLOGY 2 XET 3 ifUETandVETthenU V T 4 if Ua a6 is any collection of sets each ofwhich is in T then UWEA Ua E T A topological space is a pair X T where X is a set and T is a topology for X If X T is a topological space then U Q X is called an open set in X T if and only if U E T Theorem 29 Let Ui1 be a nite collection of open sets in a topological space X T Then 121 Ui is open Our rst step toward understanding this abstract de nition of a topo logical space is to con rm that the de nition has captured relevant features of the prototype that is the real line that spawned it So our rst example of a topological space will be the real number line where the collection of open sets in R that we talked about in the previous section is the standard topology on R Example 3 standard topology on R The standard topology Tstd for R is de ned as follows a subset U of R belongs to TQM if and only if for each point p of U there is an open interval ap bp such that p 6 ap bp C U Let us consider some other examples of topological spaces Note that X T and X T are di erent topological spaces if T 7 T even though the underlying set X is the same Keep in mind that open sets U are elements of the topology T and subsets of the space X Elements of X on the other hand are what we call the points of the space X Example 4 discrete topology For a set X let 2X be the set of all subsets of X Then T 2X is called the discrete topology on X The space X 2X is called a discrete topological space Note the spelling discrete topology not discreet topologyl Example 5 indiscrete topology For a set X T X is called the indiscrete topology for X So X 0 X is an indiscrete topological space Notice that the discrete topology has the maximum possible collection of open sets that any topology can have while the indiscrete topology has the minimum possible collection of open sets Example 6 nite complement or co nite topology For any set X the nite complement or co nite topology for X is described as follows a subset U of X is open if and only if U Q or X 7 U is nite 22 OPEN SETS AND TOPOLOGIES 15 Recall that a countable set is one that is either nite or countably in nite Example 7 countable complement topology For any set X the countable complement topology for X is described as follows a subset U of X is open if and only if U Q or X 7 U is countable Exercise 210 Verify that all the examples given above are indeed topologies in other words that they satisfy all four conditions needed to be a topology Exercise 211 1 Describe some of the open sets you get if R is endowed with the topologies described above standard discrete indiscrete co nite and countable complement Speci cally identify sets that demonstrate the differences among these topologies that is nd sets that are open in some topologies but not in others to For each of the topologies determine if the interval 01 6 R is an open set in that topology We can generalize the standard topology on R to the Euclidean spaces R Rather than using open intervals to generate open sets we use open balls Example 8 Let R be the set of all n tuples of real numbers 1 The Euclidean distance dx y between points x x1 x2 xn and y y1y2 yn is given by the equation Way imam i1 2 The open ball of radius 6 gt 0 around point p E R is the set Bx 8 90 l 1097 90 lt 6 3 A topology T for R is de ned as follows a subset U of R belongs to T if and only if for each point p of U there is a 61 gt 0 such that B map Q U This topology T is called the standard topology for R Exercise 212 Give an example of a topological space and a collection of open sets in that topological space to show that the in nite intersection of open sets need not be open 16 CHAPTER 2 GENERAL TOPOLOGY 23 Limit Points and Closed Sets As in Rstd we will de ne the concept of a limit point using open sets and then de ne closed sets as those sets that contain all their limit points De nition limit point Let X T be a topological space A be a subset of X and p be a point in X Then p is a limit point of A if and only if for each open set U containing p U 7 O A 7 0 Notice that p may or may not belong to A In other words p is a limit point of A if all open sets containing p intersect A at some point other than itself Thus the concept of a limit point gives us a way of capturing the idea of a point being arbitrarily close77 to a set without using the concept of distance Instead we use the idea of open sets in a topology De nition isolated point Let X T be a topological space A be a subset of X and p be a point in X lfp E A but p is not a limit point of A then p is an isolated point of A If p is an isolated point of A then there is an open set U such that U O A Theorem 213 Suppose p A in a topological space X T Then p is not a limit point of A if and only if there exists an open set U with p E U and U A0 Exercise 214 Give examples of a set A in a topological space and 1 a limit point of A that is an element of A to a limit point of A that is not an element of A 03 an isolated point of A q a point not in A that is not a limit point of A De nition closure of a set Let X T be a topological space and A Q X Then the closure of A denoted A or ClA is A together with all of its limit points De nition closed set Let X T be a topological space and A Q X A is closed if and only if ClA A in other words if A contains all its limit points Theorem 215 For any topological space XLT and A Q X A is closed that is for any set A in a topological space A A 23 LIMIT POINTS AND CLOSED SETS 17 A basic relationship between open sets and closed sets in a topological space is that they are complements of each other Theorem 216 Let X T be a topological space Then the set A is closed if and only if X 7 A is open Theorem 217 Let X T be a topological space and let U be an open set and A be a closed subset of X Then the set U 7 A is open and the set A 7 U is closed The properties of a topological space can be captured by focusing on closed sets instead of open sets From that perspective the four de ning properties of a topological space are captured in the following theorem about closed sets Theorem 218 Let X T be a topological space i 0 is closed ii X is closed The union of nitely many closed sets is closed iv Let A0 a6 be a collection of closed subsets in X T Then ag AD is closed Exercise 219 Give an example to show that the union of in nitely many closed sets in a topological space may be a set that is not closed Exercise 220 Give examples of topological spaces and sets in them that 1 are closed but not open 2 are open but not closed 3 are both open and closed 4 are neither open nor closed Exercise 221 State whether each of the following sets are open closed both or neither 1 ln Z with the nite complement topology 0 1 2 prime numbers n Z 10 2 In R with the standard topology 0101 0101 1n l n E N 18 CHAPTER 2 GENERAL TOPOLOGY 3 In R2 with the standard topology l x2y2 17 l x2 242 gt17 9 l 902 13 21 4 Which sets are closed in a set X with the discrete topology indiscrete topology Theorem 222 For any set A in a topological space X7 the closure of A equals the intersection of all closed sets containing A7 that is7 CIA 0 AQCCeC where C is the collection of all closed sets in X lnformally7 we can say A is the smallest closed set that contains A Exercise 223 Pick several different subsets of R and nd their closure in H the discrete topology to the indiscrete topology 03 the nite complement topology 4 the standard topology Theorem 224 Let A7 B be subsets of a topological space Then 1 AQBA Eand 2 m A U E Exercise 225 In R2 with the standard topology7 describe the limit points and closure of the following two sets 1 The topologist s sine curve sltmvsmltgtgt mlto gt 2 The topologist s comb 1 C 0 01 U 7 01 ltz Wei l g1nyiyeii The following exercise is dif cult Exercise 226 In the standard topology on R describe a non empty subset C of the closed unit interval 07 1 that is closed7 contains no non empty open interval7 and where no point of C is an isolated point Chapter 3 Separation Countability and Covering Properties At this point we know what a topology is and we have a number of ways of describing a topology eg with a basis with a total order with a topology on a larger space We now will turn our attention to properties of these topologies At the end of this chapter you will be asked to complete the following table It makes sense to give it to you now so you can ll in the properties as we go Exercise 31 Make a grid with all our examples of topologies down the side Across the top put each separation countability and covering property as we de ne it Fill in squares indicating which examples have what properties 31 Separation Properties The rst properties are the so called separation properties thus called be cause we use open sets to separate two points or closed sets from each other De nition T1 Hausdor regular normal Let X T be a topological space 1 X is T1 if and only if for all z E X is a closed set 2 X is Hausdor or T2 if and only if for each pair of points z y in X there are disjoint open sets U and V in T such that z E U and y E V 3 X is regular if and only if for each x E X and closed set A in X with z A there are open sets UV such that z E U A Q V and 25 26 CHAPTER 3 SEPARATION COUNTABILITY AND COVERING PROPERTIES U O V Q 4 X is normal if and only if for each pair of disjoint closed sets A and B in X there are open sets UV such that A Q U B Q V and U O V Q Theorem 32 Every Hausdorff space is T1 Theorem 33 Every regular T1 space is Hausdorff Theorem 34 Every normal T1 space is regular Exercise 35 Find or de ne a topological space that is not T1 Theorem 36 A topological space X is T1 if and only if for any pair of distinct points zy in X there are open sets U 9 z and V 9 y such that z V and y U Theorem 37 A topological space X is regular if and only if for each point p in X and open set U containing p there is an open set V such that p E V and V Q U Theorem 38 A topological space X is normal if and only if for each closed set A in X and open set U containing A there is an open set V such that A Q V and V Q U Theorem 39 A topological space X is normal if and only if for each pair of disjoint closed sets A and B there are disjoint open sets U and V such that Ag U B g V andU V Exercise 310 Find two disjoint closed subsets A and B of a R2 with the standard topology such that infda b l a E A and b E B O A natural question to ask is what properties carry through from a space to all of its subspaces De nition hereditary property Let P be a topological property such as T1 Hausdorff etc A topological space X is hereditarily P if and only if for each subspace Y of X Y has property P when Y is given the relative topology from X Theorem 311 A Hausdorff space is hereditarily Hausdorff Theorem 312 A regular space is hereditarily regular Theorem 313 Let A be a closed subset of a normal space X Then A is normal when given the relative topology Normality Lemma 314 Let A and B be subsets of a topological space X and let MLEN and VibeN be two collections of open sets such that 32 COUNTABILITY PROPERTIES 27 1 A g UEN U 239 B Q UieNVi7 3 foreachiinN i B andVi A Then there are open sets U and V such that A Q U B Q V and U V 0 32 Countability Properties We will now turn our attention to properties that have to do with count ability You may want to review Chapter 1 before moving ahead De nition dense Let A be a subset of a topological space X Then A is dense in X if and only ifA X De nition separable A space X is separable if and only if X has a count able dense subset Erample 12 Rstd is separable ls R not separable in any of the topologies you ve studied The choice of the word separable for the property described above is an unfortunate one given that it is not related to the separability properties we described in the previous section De nition 2nd countable A space X is 2nd countable if and only if X has a countable basis De nition neighborhood basis Let p be a point in a space X A collection of open sets Ua a6 in X is a neighborhood basis for p if and only ifp 6 U0 for each 04 E and for every open set U in X with p in U there is an 04 in such that U0 Q U De nition 15 countable A space X is 15 countable if and only if for each point z in X there is a countable neighborhood basis for m Theorem 315 A 2nd countable space is separable Theorem 316 A 2nd countable space is 1St countable Theorem 317 A 2nd countable space is hereditarily 2nd countable Theorem 318 If X is a separable Hausdorff space then le 22N Theorem 319 lfp E X and p has a countable neighborhood basis then p has a nested countable neighborhood basis 2SCHAPTER 3 SEPARATION COUNTABILITY AND COVERING PROPERTIES De nition convergence Let P pih39EN be a sequence of points in a space X Then the sequence P converges to a point x if and only if for every open set U containing z there is an integer M such that for each m gt M7 pm 6 U Theorem 320 Suppose z is a limit point of the set A in a 1St countable space X Then there is a sequence of points in A that converges to m Theorem 321 Every uncountable set in a 2nd countable space has a limit point
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