Lecture 30 - 11/2/2012

Home , Compact convergence

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 ε > 0 there is a is a ﬁnite covering of X by ε-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 ε 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 = {k|xk ∈ B1}. 1 I Inductively cover X with n -balls and choose Bn with inﬁnite intersection with Jn−1 I Let Jn = {k ∈ Jn−1|xk ∈ Bn} I Now choose nk ∈ 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 ∈ X if for all ε > 0 there is a nbdh Ux0 such that

for all x ∈ Ux0 d(f (x), f (x0)) < ε

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 ∈ X the set

Fx = {f (x)|f ∈ F}

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 ∈ X and open set U ⊂ Y let

S(x, U) = {f ∈ Y X | f (x) ∈ U}

Let S = {S(x, U)|x ∈ X and U open in Y }. 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 ∈ B if f (x1) ∈ U1, ··· , f (xn) ∈ 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 ∈ Y X a compact set K ⊂ X and ε > 0 let

X BK (f , ε) = g ∈ Y | sup{d(f (x), g(x))|x ∈ K} < ε

X Let B = {BK (f , ε)|f ∈ Y , K ⊂ X compact and ε > 0}. 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 fn|K 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 |K 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 |K 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 ∈ C(X , Y ) | f (C) ⊂ U

Let S = {S(K, U)|K ⊂ X compact and U open in Y }. 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