MA4266 Topology Lecture 16 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 Separation Properties or axioms, specify the degree by which points and/or closed sets can be separated by open sets & continuous functions T0 Kolmogorov space Ex. Sierpinski space 1 point from a pair separated from the other by OS T1 Frechet space Ex. Finite Comp. Top. on Z each point from a pair separated from the other by OS ~ 2 T2 Hausdorf space Ex. Double Origin Top. on R { 0 } pairs of points jointly separated by OS T2 1 completely Hausdorf space (called Urysohn in book) 2 T3 PP sep. by CN Ex. Half-Disc Top. on R [0, ) points & closed sets jointly separated by OS Ex. Tychonov & Hewitt & Thomas’s Corkscrew Top., Ostaszewski T3 1 points & closed sets jointly separated by CF 2 Ex. Sorgenfrey plane T4 pairs of closed sets jointly separated by OS Combinations of Separation Properties Definition A space X is Completely Hausdorff or T2 1 2 a, b X , a b openU a,V b,U V . Regular if it is T1 and T3 , Completely Regular or T3 1 2 if it is T1 and a X , a closedC X if continuous f:X R, f (a) 0, f (C ) 1, T1 and T4 . Theorem 8.1 T1 finite subsets are closed. Normal if it is Metrizable Normal Completely Regular Regular T2 1 T2 T1 T0 2 Theorem 8.2 Products of Ti spaces are Ti , i 0,1, 2. Regular Spaces X is a T1 space. Then X is T3 (and therefore regular) if and only if for every a X and open U a there exists open W a with W U . Proof If X is regular and a U open X then C X \ U is closed and a C hence there exist disjoint open W a and V C. Hence W X \V . Theorem 8.3 Assume that W X \ V (why?) so W X \ C U . Conversely, if the latter condition holds and C is a closed set with a C. Then there exists open W a with W X \ C (why?) so W and X \ W are disjoint open sets containing a and C respectively. Hence Regular Spaces X is a T1 space. Then X is T3 if and only if for every a X , a C closed, there exists open U , V with a U , C V ,U V . Theorem 8.4 Assume that Proof page 235. Theorem 8.5 The product of regular spaces is regular. Proof Let { X : A} be a family of regular spaces, X A X , a U open X. Therefore n 1 a p Ui open X , i 1,...,n i1 i (Ui ) U . Then pi (a) Vi open Vi Ui , i 1,...,n why? n 1 V p Then i1 i (Vi ) is open, contains n 1 n V i 1 pi (Vi ) i 1 pi1 (Ui ) U , so X a and is regular. Examples Double Origin Topology (counterexample # 74, [1]) ( X , ), X ( R \ {0}) {0 } { 0 } 2 (R \ {0})usual 0 0 2 0 has a local basis Vn, { ( x, y) : x y n12 , y 0} {0 }, n N Question Why is ( X , ) T2 ? Question Why is ( X , ) NOT T2 1 ? 2 2 2 ( X , ) 2nd countable ? path connected ? Question Is ( X , ) regular ? locally compact ? Question Is [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 8.2.1 in Croom’s Principles of Topology. ( X , ), X R (0, ) R {0} ( R (0, ))usual ( R {0}) half disk where a local basis at (a,0) R {0} is Vn { ( x, y) : ( x a) y n12 , y 0} {(a,0)}, n N 2 Question Why is 2 ( X , ) T2 1 ? 2 Question Why is ( X , ) NOT T3 ? Normal Spaces X is a T1 space. Then X is T4 (and hence normal) iff for every closed A X and open U A there exists open W A with W U . Theorem 8.6 Assume that X is a T1 space. Then X is T4 iff for every pair of disjoint closed sets A, B X there exist open sets U A, V B with U V . Theorem 8.7 Assume that Theorem 8.8 Every compact Hausdorff space is normal. Proof Corollary to Theorem 6.5, pages 165-166. Normal Spaces Theorem 8.9 Every regular Lindelöf space is normal. A, B be disjoint closed sets. First, use regularity to construct an open cover of A by sets whose closures are disjoint with A, likewise for B. Proof Let Second, use the Lindelöf property to obtain countable subcovers U1 ,U 2 ,... of A and V1 ,V2 ,... of B. n n Third, construct U U n \ Vn , V Vn \ U n i 1 i 1 ' n and observe that ' n U V , for all m, n N. ' n Fourth, construct U ' m ' U , V V n1 n1 n and ' n observe they are open sets and A U , B V . Normal Spaces Why ? Corollary Every 2nd countable regular space is normal. Definition For a set A, card A c A is equipotentwith [0,1] X is a separable normal space and E card E c, then E has a limit point. Proof Assume that such a set E has no limit point. Then for every Y E the sets Y and E \ Y are closed so there exist disjoint open UY Y and VY E \ Y . Let D Be a countable dense subset and construct a function h : P( E ) P( D) by h(Y ) UY D, Y P( E). Theorem 8.10 If is a subset with Since h is 1-to-1 (see p. 239) card P( E) card P( D). But card P( D) card R card P(R) card P(E ). Theorem 8.11 Every metric space is normal. Ex 3.2 p.69 Examples Sorgenfrey Plane (counterexample # 84, [1]) and Example 8.3.1 in Croom’s Principles of Topology. ( R R, ) half openintervaltopology Question Why is ( R, ) regular ? Question Why is ( R, ) Lindelöf ? Question Why is ( R, ) normal ? Question Why is ( R R, ) regular, separable ? Let E {( x, x) : x R} R R Question What is the subspace topology on E ? Question What are the limit points of E ? Question Why is ( R R, ) NOT normal ? Examples Niemytzki’s Tangent Disc Top. (counterexample # 82,[1]) and Ex. 8.3, Q6, p. 242 Croom’s Principles of Topology. ( X , ), X R (0, ) R {0} (R (0, ))usual (R {0}) tangentdisk where a local basis at (a,0) R {0} Vn { ( x, y) : ( x a) ( y ) n12 } {( x,0)}, n N 2 Question Why is Question Why is Question Why is 1 2 n ( X , ) T3 1 ? 2 ( X , ) separable ? ( X , ) NOT normal ? Separation by Continuous Functions Definition Separation by continuous functions. and Ex. 8.4.1, Q6, p. 243 Croom’s Principles of Top. Theorem 8.12 Let X be a T1 space. (a) If points a and b can be separated by a continuous 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 : ( X1 ,1 ), X1 R, 1 ( R \ {0})discrete {0} where U 0 is open iff R \ U is finite. (a one-point compactification of an uncountable set) Definition A subset S of a topological space X is a G set (gee-delta) if it is the intersection of a countable collection of open sets, and a F set (eff-sigma) if it is the union of a countable collection of closed sets. Theorem If X is a topological space and f : X R is 1 f (a) is a G set for every a X . continuous then 1 Proof f (a) f 1 i 1 B(a, ) i 1 f 1 ( B(a, 1i )). 1 i Corollary Every continuous f : X1 R equals f (0) except at a countable set of points. Examples Thomas’ Plank (counterexample # 93, [1]) ( X , ), X X1 X 2 \ {(0,0)}, prod top 1 ( X , ), X { where 2 2 2 k : k Z \ {0}} {0} 2 usual subspace topologyof R Theorem If f : X R is continuous then f is constant except at a countable set of points. R { } the function f is constant 1 countable. R { } R \ R on a set k where k k 1 L kZ \{0} Rk L { f is constant on each k } where Proof On each set and therefore 1 k f is constant on ( L \ {0}) {0}. Examples Thomas’ Corkscrew (counterexample # 94, [1]) (Y , ),Y X Z {a } {a } where the local bases for points in Y \ {( x,0, m) : x 0, m Z} is the same as for the product topology, and local bases for other points are Bn ( x,0, m) {( x,0, m)}{ ( x, 1k , m) : k n} { ( x, 1k , m 1) : k n} Bn (a ) {a } m n X {m}, Bn (a ) {a } m n X {m}, for n N. Theorem (Y , ) is regular but NOT completely regular since every continuous f : Y R satisfies f (a ) f (a ). Separation by Continuous Functions Lemma 1. Dyadic numbers are dense in R. Lemma 2. Let X be a space and D dense [0, ). t D openUt X (a) t1 t2 Ut Ut , and (b) tD U t X , 1 2 then the function f : X R defined by f ( x) glb {t D : x Ut }, x X is continuous. Theorem 8.13 Urysohn’s Lemma Let X be a T1 space. then X is normal iff for all disjoint closed A, B X , If for every there exists a continuous f : X R with f ( A) 0, f ( B) 1. Theorem 8.14 Tietze Extension Theorem Let X be a normal space, A closed X , and f : A R continuous. Then f has a continuous extension F : X R. Assignment 16 Read pages 234-237, 237-241, 243-251 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