MA4266 Topology Wayne Lawton Department of Mathematics S , Lecture 10

Covers Definition: Letbe a subset of a set A cover ofis a collection of subsets of such that Examples Question Which covers are finite ? Countable ?

Subcovers Definition: Letbe a subset of a set a cover of such that A subcover (ofderived from and ) is a ofcover be Lemma Ifis finite then every cover ofhas a finite subcover. Question Give an example of this lemma.

Open Covers Definition: Letbe a subset of a topological space An open cover ofis a cover of open. whose elements are Example is an open cover and is an subcover (obviously this subcover is open) An finite cover ofis a cover ofthat is finite.

Compact Spaces Definition: A topological spaceis compact if every open cover of of has a finite subcover. equivalent conditions: 1.is a compact topological space (regarded as A subspace is compact if it satisfies either of the following a topological space with the subspace topology). 2. Every open cover ofthat consists of open subsets ofhas a finite subcover.

Examples Example (of compact spaces) (a) Finite spaces with the finite complement topology (b) Closed bounded intervals (c) Closed bounded subsets of (d) Example (of noncompact spaces) (a) Any infinite discrete topological space (b) The open interval (c)

Finite Intersection Property Definition A family of subsets of X has the finite Question What is the intersection of (all) these sets? has nonempty intersection. Example Theorem 6.1: A spaceis compact iff every family of intersection property if every finite subcollection closed sets in has nonempty intersection. with the finite intersection property Proof Follows from De Morgan’s Laws.

Cantor’s Theorem of Deduction Theorem 6.2: Let Proof ??? sequence of nonempty, closed, bounded subsets of Then Question Why assume both closed and bounded ? be a nested Theorem 6.3: Closed subset of compact  compact. Proof ???

Properties of Compact Sets be a compact subset of a Hausdorff spaceProof Let and let Theorem 6.4: Compact subset of Hausdorff  closed. It suffices to show that there exists an open subsetwith Sinceis Hausdorff, for everythere exist disjoint open subsetswith Clearlyis an open cover of Sinceis compact there exist such thatis a cover ofConstruct Then is open.and

Properties of Compact Sets and of a Hausdorff spacethen there exist disjoint open Theorem 6.5: Ifare disjoint compact subsets subsetswithandof Proof Similar to the proof of theorem 6.4. Question Construct an example of a compact subset of a topological spacesuch thatis not closed. Hint:has two points.

Compactness and Continuity space is compact. Theorem 6.6: The continuous image of a compact Proof Easy. is continuous then is compact andis Hausdorff and is closed. Theorem 6.7: If Theorem 6.8: Ifis compact andis Hausdorff and is a continuous bijection thenis a homeomorphism.Proof Easy. Theorem 6.9: Ifis compact and is continuous then there exists such that Proof Easy.

Compactness and Continuity be metric spaces. ADefinition Let function there exists is uniformly continuous if for every such that Theorem 6.10: Ifis a compact metric space isandis a metric space and continuous thenis uniformly continuous. Proof see pages Question Show that uniform continuity  continuity. Question Give an example of a continuous function that is not uniformly continuous.

Tutorial Assignment 10 Read pages Exercise 6.1 problems 3, 6 Exercise 6.2 problems 1, 2, 3, 4 Complete the proof of the Lemma, page 172 for n > 2.