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MA4266 Topology

Lecture 11. MA4266 Topology. Wayne Lawton Department of Mathematics S17-08-17, 65162749 matwml@nus.edu.sg 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. is compact iff it is. closed and bounded.

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MA4266 Topology

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  1. Lecture 11 MA4266 Topology Wayne Lawton Department of Mathematics S17-08-17, 65162749 matwml@nus.edu.sg http://www.math.nus.edu.sg/~matwml/ http://arxiv.org/find/math/1/au:+Lawton_W/0/1/0/all/0/1

  2. Basics Theorem 6.11: A subset of 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

  3. 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 such that Therefore is an open cover of B. Furthermore 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.

  4. 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 in the Hilbert space does not have the BWP (why?).

  5. Lebesgue Number of an Open Cover Definition: Let be a metric space and an open cover of A Lebesgue number for such that every subset of is a positive number having diameter less than is contained in some element in Theorem 6.16 If is a compact metric space has a Lebesgue number. then every open cover of Proof follows from the following Lemma 1 since each subset having diameter less than is a subset of an open ball of radius

  6. BWExistence of Lebesgue Number Lemma 1: Let be a metric space that satisfies the Bolzano-Weierstrass property. Then every open cover of has a Lebesgue number. and assume to the Proof Let be an open cover of contrary that does not have a Lebesgue number. in such that Then there exists a sequence for every and for every Then is infinite (why?) so the BW property implies that it has a limit point so there exists contains and with Then infinitely many members of

  7. BWExistence of Lebesgue Number Hence contains some with Then for so This contradicting the initial assumption that for all and completes the proof of Lemma 1.

  8. Total Boundedness Definition: Let be a metric space and An net for is a finite subset such that The metric space is totally bounded if it has an net for every Lemma 2: Let be a metric space that satisfies the Bolzano-Weierstrass property. Then is TB. Proof Assume to the contrary that there exists such that does not have an net. Choose and construct a sequence with that has no limit point.

  9. Compactness and the BWP Theorem 6.15: For metric spaces compactness = BWP. Proof Theorem 4.14 implies that compactness  BWP. For the converse let be an open cover of a metric space having the Bolzano-Weierstrass property. such that for Lemma 1 implies that there exists every the open ball is contained in some member of Lemma 2 implies that there exists a finite subset such that an open cover of Choose and observe that covers

  10. Compactness for Subsets of the following Theorem 6.17: For a subset conditions are equivalent: (a) is compact. (b) has the BWP. (c) is countably compact. (d) is closed and bounded. Question Are these conditions equivalent for

  11. Assignment 11 Read pages 175-180 Exercise 6.3 problems 2, 3, 4, 5, 9, 13, 14, 15

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