Lecture 30 - 11/2/2012 Math 5801 General Topology and Knot Theory Nathan Broaddus Ohio State University November 2, 2012 Nathan Broaddus General Topology and Knot Theory Lecture 30 - 11/2/2012 Compact Metric Spaces Two generalizations of pointwise convergence Course Info Reading for Monday, November 5 Chapter 7.46, pgs. 172-177 HW 10 for Monday, November 5 I Chapter 2.24: 3, 5a-d, 8a-d, 12a-f (see pg. 66 for required definitions) I Chapter 2.25: 1, 2a-c Midterm 2 Friday, November 9 I Munkres Chapters 2-3, 7 Nathan Broaddus General Topology and Knot Theory Lecture 30 - 11/2/2012 Compact Metric Spaces Two generalizations of pointwise convergence Compact Metric Spaces Definition 244 (Totally Bounded) A metric space (X ; d) is totally bounded if for every " > 0 there is a is a finite covering of X by "-balls. Proposition 245 (Compact metric spaces are totally bounded) A metric space is compact if and only if it is complete and totally bounded. Proof. I Suppose (X ; d) is compact. I Then X is sequentially compact. I Every sequence in X has a convergent subsequence. I Hence Cauchy sequences have convergent subsequences. I Hence X is complete. I Total boundedness follows from fact that X has an open cover by " balls. Nathan Broaddus General Topology and Knot Theory Lecture 30 - 11/2/2012 Compact Metric Spaces Two generalizations of pointwise convergence Compact Metric Spaces Proof of Prop. 245 (continued). I Suppose (X ; d) is complete and totally bounded. I Let (xn) be a sequence in X . I Cover X by finitely many 1-balls I At least one ball B1 has infinitely many xn's I Let J1 = fkjxk 2 B1g. 1 I Inductively cover X with n -balls and choose Bn with infinite intersection with Jn−1 I Let Jn = fk 2 Jn−1jxk 2 Bng I Now choose nk 2 Jk so that nk < nk+1 I The subsequence (xnk ) is Cauchy. I By completeness of X this Cauchy sequence convereges. Nathan Broaddus General Topology and Knot Theory Lecture 30 - 11/2/2012 Compact Metric Spaces Two generalizations of pointwise convergence Compact Metric Spaces Definition 246 (Equicontinuous) Let (Y ; d) be a metric space and F ⊂ C(X ; Y ) be a set of maps. F is equicontinuous at x0 2 X if for all " > 0 there is a nbdh Ux0 such that for all x 2 Ux0 d(f (x); f (x0)) < " Definition 247 (Pointwise bouned) Let (Y ; d) be a metric space and F ⊂ C(X ; Y ) be a set of maps. F is pointwise bounded if for each x 2 X the set Fx = ff (x)jf 2 Fg is bounded in Y . Nathan Broaddus General Topology and Knot Theory Lecture 30 - 11/2/2012 Compact Metric Spaces Two generalizations of pointwise convergence Compact Metric Spaces Proposition 248 (Ascoli's Theorem) Let (Rn; d) be euclidean n-space X a compact space. A set of maps F ⊂ C(X ; Rn) has compact closure in the uniform topology if and only if it is equicontinuous and pointwise bounded. Nathan Broaddus General Topology and Knot Theory Lecture 30 - 11/2/2012 Compact Metric Spaces Two generalizations of pointwise convergence Two generalizations of pointwise convergence Definition 249 (Point-open topology) For a point x 2 X and open set U ⊂ Y let S(x; U) = ff 2 Y X j f (x) 2 Ug Let S = fS(x; U)jx 2 X and U open in Y g. The pointwise convergence or point-open topology on Y X is the topology generated by the subbasis S. I Typical basis element in point-open topology is B = S(x1; U1) \ S(x2; U2) \···\ S(xn; Un) I f 2 B if f (x1) 2 U1; ··· ; f (xn) 2 Un Proposition 250 A sequence (fn) converges in the point-open topology iff it converges pointwise. Nathan Broaddus General Topology and Knot Theory Lecture 30 - 11/2/2012 Compact Metric Spaces Two generalizations of pointwise convergence Two generalizations of pointwise convergence Definition 251 (Compact convergence topology) (Y ; d) a metric space and X a space. For a point f 2 Y X a compact set K ⊂ X and " > 0 let X BK (f ;") = g 2 Y j supfd(f (x); g(x))jx 2 Kg < " X Let B = fBK (f ;")jf 2 Y ; K ⊂ X compact and " > 0g. The compact convergence topology on Y X is the topology generated by the basis B. Proposition 252 A sequence (fn) converges to f in the compact convergence topology iff for each compact set K ⊂ X the sequence fnjK converges uniformly to fK . Nathan Broaddus General Topology and Knot Theory Lecture 30 - 11/2/2012 Compact Metric Spaces Two generalizations of pointwise convergence Two generalizations of pointwise convergence Definition 253 (Compactly generated) X is compactly generated if a subset A ⊂ X and for every compact K ⊂ X we have K \ A is open in K then A is open in X . Lemma 254 Let X be compactly generated. Then a function f : X ! Y is continuous if and only if f jK is continuous for each compact K ⊂ X . Nathan Broaddus General Topology and Knot Theory Lecture 30 - 11/2/2012 Compact Metric Spaces Two generalizations of pointwise convergence Two generalizations of pointwise convergence Proposition 255 If X is compactly generated and Y is a metric space then C(X ; Y ) is closed in Y X in the compact convergence topology. Proof. I Let fn ! f . 1 I Then for K ⊂ X compact BK (f ; n ) shows fn ! f uniformly I Hence f jK is continuous. Nathan Broaddus General Topology and Knot Theory Lecture 30 - 11/2/2012 Compact Metric Spaces Two generalizations of pointwise convergence Two generalizations of pointwise convergence Definition 256 (Compact-open topology) Let X and Y be spaces. For compact K ⊂ X and open U ⊂ Y let S(K; U) = f 2 C(X ; Y ) j f (C) ⊂ U Let S = fS(K; U)jK ⊂ X compact and U open in Y g. The compact-open topology on C(X ; Y ) is the topology generated by the subbasis S. Proposition 257 Let X be a space and Y be a metric space. Then the compact convergence and compact-open topologies coincide on C(X ; Y ). Nathan Broaddus General Topology and Knot Theory Lecture 30 - 11/2/2012 Compact Metric Spaces Two generalizations of pointwise convergence Two generalizations of pointwise convergence Proposition 258 Let X be locally compact Hausdorff and let C(X ; Y ) have the compact-open topology. Then the evaluation map e : X × C(X ; Y ) ! Y given by e(x; f ) = f (x) is continuous. Proposition 259 Let X and Y spaces and give C(X ; Y ) the compact-open topology. If f : X × Z ! Y is continuous then the induced function F : Z ! C(X ; Y ) given by F (z)(x) = f (x; z) is continuous. If X is locally compact Hausdorff then the converse holds. Nathan Broaddus General Topology and Knot Theory.