MA4266 Topology Wayne Lawton Department of Mathematics S , Lecture 16. Tuesday 6 April 2010

Separation Properties or axioms, specify the degree by which points and/or closed sets 1 point from a pair separated from the other by OS Kolmogorov space can be separated by open sets & continuous functions Frechet space each point from a pair separated from the other by OS Hausdorf space pairs of points jointly separated by OS completely Hausdorf space (called Urysohn in book) points & closed sets jointly separated by OS points & closed sets jointly separated by CF pairs of closed sets jointly separated by OS Ex. Sierpinski space Ex. Finite Comp. Top. on Z Ex. Double Origin Top. on PP sep. by CN Ex. Half-Disc Top. on Ex. Tychonov & Hewitt & Thomas’s Corkscrew Top., Ostaszewski Ex. Sorgenfrey plane

Combinations of Separation Properties Regular if it isand Metrizable  Normal  Completely Regular  Regular  Normal if it is Theorem 8.1 finite subsets are closed. Theorem 8.2 Products ofspaces are Completely Hausdorff orDefinition A spaceis if  Completely Regular or if it isand

Regular Spaces Theorem 8.3 Assume thatis aspace.Then is(and therefore regular) if and only if for every and openthere exists openwith Proof Ifis regular and then is closed andhence there exist disjoint openHence (why?)so Conversely, if the latter condition holds andis a closed set withThen there exists openwith (why?) soandare disjoint open sets containingandrespectively.

Regular Spaces Theorem 8.4 Assume thatis aspace.Then isif and only if for every there exists openwith Proof page 235. why? Then be a family of regular spaces, Therefore is open, contains Then Theorem 8.5 The product of regular spaces is regular. Proof Let and sois regular.

Examples Double Origin Topology (counterexample # 74, [1]) NOT has a local basis Question Why is Question Is2 nd countable ? path connected ? Question Isregular ? locally compact ? [1] Counterexamples in Topology by Lynn Arthur Steen and J. Arthur Seebach, Jr., Dover, New York, 1970.

Examples Half-Disc Topology (counterexample # 78, [1]) and Example in Croom’s Principles of Topology. where a local basis atis Question Why is NOTQuestion Why is

Normal Spaces Theorem 8.6 Assume thatis aspace.Then is iff for every pair of disjoint closed sets openthere exists openwith there exist open setswith Theorem 8.8 Every compact Hausdorff space is normal. Proof Corollary to Theorem 6.5, pages and Theorem 8.7 Assume thatis aspace.Then is (and hence normal) iff for every closed

Normal Spaces Theorem 8.9 Every regular Lindelöf space is normal. Fourth, construct Proof Let likewise for First, usebe disjoint closed sets. subcoversofand Third, construct regularity to construct an open cover of whose closures are disjoint with by sets Second, use the Lindelöf property to obtain countable and observe that of observe they are open sets and and

Normal Spaces Why ? Corollary Every 2 nd countable regular space is normal. Definition For a set Theorem 8.10 Ifis a separable normal space and is a subset withthenhas a limit point. Proof Assume that such a sethas no limit point. Then for everythe setsandare closed so there exist disjoint openand function LetBe a countable dense subset and construct a by Sinceis 1-to-1 (see p. 239) But Theorem 8.11 Every metric space is normal. Ex 3.2 p.69

Examples Sorgenfrey Plane (counterexample # 84, [1]) and Example in Croom’s Principles of Topology. Question Why isregular ? Let Question Why is Lindelöf ? Question What is the subspace topology on Question What are the limit points of Question Why isNOT normal ? Question Why is normal ? Question Why is regular, separable ?

Examples Niemytzki’s Tangent Disc Top. (counterexample # 82,[1]) and Ex. 8.3, Q6, p. 242 Croom’s Principles of Topology. Question Why is NOT normal ? where a local basis at Question Why isseparable ? Question Why is

Separation by Continuous Functions Definition Separation by continuous functions. and Ex , Q6, p. 243 Croom’s Principles of Top. Theorem 8.12 Let (a) If points a and b can be separated by a continuous be aspace. function then they can be separated by open sets. (b) If each point x and closed set C not containing a can be separated by continuous functions then they can be separated by open sets. (c) If disjoint closed sets A and B can be separated continuous functions then they can be separated by open sets.

Examples Definition Funny Line : whereis open iffis finite. Theorem If (a one-point compactification of an uncountable set) Definition A subset S of a topological space X is a set (gee-delta) if it is the intersection of a countable collection of open sets, and a set (eff-sigma) if it is the union of a countable collection of closed sets. is a topological space and continuous thenis aset for every is Proof Corollary Every continuousequals except at a countable set of points.

Examples Thomas’ Plank (counterexample # 93, [1]) Theorem Ifis continuous then except at a countable set of points. is constant Proof On each setthe function on a setwhere countable. is constant is constant on eachwhere is constant onand therefore where

Examples Thomas’ Corkscrew (counterexample # 94, [1]) is the same as for the product topology,and local bases where the local bases for points in for Theorem for other points are is regular but NOT completely regular since every continuoussatisfies

Separation by Continuous Functions Lemma 1. Dyadic numbers are dense in Lemma 2. Letbe a space and If for every (a)and(b) then the functiondefined by is continuous. Theorem 8.13 Urysohn’s Lemma Letbe aspace. thenis normal iff for all disjoint closed there exists a continuouswith Theorem 8.14 Tietze Extension Theorem Letbe a normal space,andcontinuous. Thenhas a continuous extension

Assignment 16 Read pages , , Prepare to solve during Tutorial Thursday 8 April Exercise 8.2 problem 4 (c) Exercise 8.3 problem 6 (a),(b),(c),(d) Exercise 8.4 problems 8 (a),(b), 11, 13, 14 (a),(b) 15, 16