MA4266 Topology Wayne Lawton Department of Mathematics S , Lecture 11. Friday 5 March 2010

Basics Theorem 6.11: A subset ofis compact iff it is Definition: A topological space is countably compact closed and bounded. 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 such thatTherefore 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.

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

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

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

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

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

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

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

Assignment 11 Read pages Prepare to solve on the board Tuesday 9 March Exercise 6.3 problems 2, 3, 4, 5, 9, 13, 14, 15

Definition: A compact, connected, locally connected Supplementary Materials Metric space is called a Peano space (or P. continuum). Examples: closed balls in Theorem (Hahn-Mazurkiewicz): A topological space is a Peano space iff it is Hausdorff and there exists a continuous surjection