#### Transcript MA4266_Lect11 - Department of Mathematics

MA4266 Topology Lecture 11. Friday 5 March 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 Basics Theorem 6.11: A subset of R n is compact iff it is closed and bounded. Definition: A topological space is countably compact if every countable open cover has a finite subcover. Definition: A topological space is a Lindelöf space if every cover has a countable subcover. Theorem 6.12: If X is a Lindelöf space, then X is compact iff it is countably compact. Theorem 6.13: The Lindelöf Theorem Every second countable space is Lindelöf. Proof see page 175 Bolzano-Weierstrass Property Definition: A topological space X has the BW-property if every infinite subset of X has a limit point. Theorem 2.14: Every compact space has the BWP. Proof Assume to the contrary that X is a compact space and that B is an infinite subset of X that has no limit points. Then B is closed (why?) and B is compact (why?). Since B has no limit points, for every point x in B there exists an open set O X such that O B {x}. Therefore x x O {Ox : x B} is an open cover of B. Furthermore O does not have a finite subcover of B (why?). Definition p is an isolated point if {p} is open. See Problem 10 on page 186. Examples Example 6.3.1 (a) Closed bounded intervals [a,b] have the BWP. (b) Open intervals do not have the BWP. (c) Unbounded subsets of R do not have the BWP. (d) The unit sphere S in the Hilbert space 2 (N ) N {1,2,3,4,...} ( N ) { ( x1 , x2 ,...) : xk R, || x || k 1 x } 2 S { x ( N ) : || x || 1 } 2 does not have the BWP (why?). 2 2 k Lebesgue Number of an Open Cover ( X , d ) be a metric space and O an open cover of X . A Lebesgue number for O Definition: Let is a positive number such that every subset of X having diameter less than element in is contained in some O. Theorem 6.16 If (X ,d) is a compact metric space then every open cover of X has a Lebesgue number. Proof follows from the following Lemma 1 since each subset having diameter less than is a subset of an open ball of radius . BWExistence of Lebesgue Number Lemma 1: Let (X ,d) be a metric space that satisfies the Bolzano-Weierstrass property. Then every open cover O Proof Let of X has a Lebesgue number. O contrary that be an open cover of X and assume to the O does not have a Lebesgue number. in X such that Then there exists a sequence {x } n n 1 B( xn , 1n ) O for every n and for every O O. Then {xn }n 1 is infinite (why?) so the BW property implies that it has a limit point a so there exists 0 and O O with B ( a, ) O. Then B(a, 2 ) contains infinitely many members of {xn }n 1. BWExistence of Lebesgue Number xn with B ( a , Hence 2 ) contains some Then for z B( xn , 1n ) 1 n 2 . d (a, z ) d (a, xn ) d ( xn , z ) 2 2 so B( xn , 1n ) B(a, ) O. This contradicting the initial assumption that for all B( xn , 1n ) O, n 1, O O and completes the proof of Lemma 1. Total Boundedness ( X , d ) be a metric space and 0. An net for X is a finite subset A X such that d ( x, A ) , x X . The metric space X is totally bounded if it has an net for every 0. Definition: Let Lemma 2: Let ( X , d ) be a metric space that satisfies the Bolzano-Weierstrass property. Then X is TB. Proof Assume to the contrary that there exists such that X does not have an net. Choose a1 X and construct a sequence {ak }k 1 with ak 1 j 1 B (a j , ) that has no limit point. k 0 Compactness and the BWP Theorem 6.15: For metric spaces compactness = BWP. Proof Theorem 4.14 implies that compactness BWP. For the converse let space O be an open cover of a metric ( X , d ) having the Bolzano-Weierstrass property. Lemma 1 implies that there exists 0 such that for x X the open ball B( x, ) is contained in some member of O. Lemma 2 implies that there exists a finite every subset A {x1 ,..., xn } X such that {B( xk , ) : k Choose B( xk , ) Ok O, k and observe that {Ok : k 1,..., n} covers X . an open cover of X. 1,..., n} 1,..., n Compactness for Subsets of R Theorem 6.17: For a subset A R n n the following conditions are equivalent: is compact. (c) A A A (d) A is closed and bounded. (a) (b) has the BWP. is countably compact. Question Are these conditions equivalent for A 2 ( N ) ? Assignment 11 Read pages 175-180 Prepare to solve on the board Tuesday 9 March Exercise 6.3 problems 2, 3, 4, 5, 9, 13, 14, 15 Supplementary Materials Definition: A compact, connected, locally connected Metric space is called a Peano space (or P. continuum). Examples: closed balls in R n Theorem (Hahn-Mazurkiewicz): A topological space X is a Peano space iff it is Hausdorff and there exists a continuous surjection f : [0,1] X . http://en.wikipedia.org/wiki/Space-filling_curve#The_Hahn-Mazurkiewicz_theorem