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MA4266 Topology Lecture 8. Friday 12 Feb 2010 Wayne Lawton Department of Mathematics S17-08-17, 65162749 [email protected] http://www.math.nus.edu.sg/~matwml/ http://arxiv.org/find/math/1/au:+Lawton_W/0/1/0/all/0/1 Separation Definition Let (X,T) be a topological space. A separation of X is are elements A, B in T with A B A , B A B X Connected and Disconnected Spaces Definition A topological space (X,T) is disconnected if it admits a separation, otherwise it is connected. Example 5.1.1 A discrete space with > 1 point is disconnected { p, q} X A { p}, B X \ A gives separation A trivial space is connected R \ {0} is disconnected R \ R {0} is disconnected 2 R {0} {(0,1)}, F {(0,1)} : F R {0} is connected Real Numbers with Usual Topology Example 5.1.2 It is CONNECTED ! Proof Assume to the contrary that there exists a separation A, B for R and choose a A, b B. Define A0 A [a, b]. Since A0 is nonempty and bounded above (why?) it has a least upper bound c. Then c A0 (why?) so c b (why?) Therefore (c, b] B (why?). Since B is closed c B. This leads to a contradiction c B A . A Subset Y of Topological Space (X,T) is called disconnected if the subspace Y is disconnected, this means that there are elements A, B in T with A B Y A Y , B Y Y A B Examples [0,1] [2,3] ( R, Tusual ) is disconnected Proof Choose A (1, 32 ), B ( 32 ,4) Q ( R, Tusual ) is disconnected Proof Choose A (, 2 ), B ( 2 , ) Separated Sets Definition Let (X,T) be a topological space. Nonempty A, B X are separated if A B A B Theorem 5.1 The following statements are equivalent 1. X is disconnected 2. X is the union of two disjoint, non-empty closed sets 3. X is the union of two separated sets 4. There exists a continuous function f:X { a, b }_disc. 5. X has a proper subset that is both open and closed 6. X has a proper subset A such that A X \ A Connectedness is a Continuous Invariant Theorem 5.2 Let X be a connected space and Y be a space and f : X Y be continuous. Then Y is connected. Proof Assume to the contrary that Y is disconnected. Then Y has a separation A, B. Then the sets U f 1 ( A), V f 1 ( B) are (a) open (why?), (b) disjoint (why?), (c) nonempty (why?), (d) have union X (why?). What does this say about these sets? Are we done? Closure of Connected is a Connected Theorem 5.4 If Y is a connected subset of a space X then Y is connected. Proof (see page 136 for a different proof). Assume to the contrary that Y is disconnected. Then there exists open subsets A, B X A Y , B Y are a separation of Y . Then A Y , B Y (why?). Therefore A Y , B Y (why?). Hence A Y , B Y are a separation of Y . such that What does this imply about Y ? Are we done ? Example 5.2.3 Topologists Sine Curve Unions of Connected Sets Theorem 5.5 Let X be a space and { A : I } a family of subsets with W I A . Then Y I A is connected. U ,V be open subsets of X such that U Y , U V Y , Y U V . It suffices to show that V Y . Choose b W and choose I so that U A . Since Proof Let A is connected Since each so Y U A A U is connected each and hence b U . A U and hence V Y U V Y . Components Definition A component of a topological space X C X which is not a proper subset of any connected subset of X . is a connected subset Theorem If X is a topological space then the relation a b if there exists a connected subset C X such that a, b C is an equivalence relation. Proof Reflexivity and symmetry are obvious, and transitivity follows theorem 5.5. Corollary Each equivalence class is a component. Corollary Each component is closed. If there are a finite number of components, they are also closed. Connected Subsets of R Theorem 5.7 R C , C connected C is an interval. Proof Clearly a C , b C , a c [a, b] C. Hence if C is not bounded above then a C [a, ) C and if C is not bounded below then a C (, a] C. So if C is not bounded above or below then C = R. If R is bounded below but not above then either C [a, ) if a inf( C ) C or C (a, ) if a inf( C ) C and similarly for 6 cases where R is bounded above. Theorem 5.8 The Intermediate Value Theorem Let f : [a, b] R be continuous and y 0 be between f (a ) and f (b ). Then there exists c [ a, b] with f (c) y0 . Assignment 8 Read pages 131-146 Do Exercise 5.1 problem 3 Prove the six properties after the definition on page 139 Do Exercise 5.2 problems 5, 8, 14, 16 Do Exercise 5.3 problem 3 Do Exercise 5.4 problems 1, 2