Advanced Calculus (II)

Advanced Calculus (II)

W^EN-C^HING L^IEN

Department of Mathematics National Cheng Kung University

2009

WEN-CHINGLIEN Advanced Calculus (II)

Ch9: Convergence in R

9.4: Compact Sets

Definition (9.35)

LetV = {Vα}α∈A be a collection of subsets of Rⁿ, and suppose that E ⊆Rⁿ.

(i)V is said to cover E (or be a covering of E) if and only if E ⊆ [

α∈A

Vα.

(ii)V is said to be an open covering of E if and only ifV covers E and each V_α is open.

(iii) LetV be a covering of E. V is said to have a finite (respectively, countable) subcovering if and only if there is a finite (respectively, an at most countable) subset A0 of A such that{Vα}α∈A0 covers E.

WEN-CHINGLIEN Advanced Calculus (II)

Theorem (9.39 Lindel ¨of)

Let n ∈N and let E be a subst of Rⁿ. If{Vα}α∈A is a collection of open sets and E ⊆ ∪α∈AVα, then there is an at most countable subset A0 of A such that

E ⊆ [

α∈A0

Vα.

Proof.

LetT be the collection of open balls with rational radii and rational centers, i.e., centers that belong to Qⁿ.This

collection is countable. Moreover, by the proof of the Borel Covering Lemma, T ”approximates” the collection of open balls in the following sense: Given any open ball

Br(x) ⊆Rⁿ, there is a ball B_ρ(a) ∈ T such that x∈B_ρ(a) and Bρ(a) ⊆Br(x).

WEN-CHINGLIEN Advanced Calculus (II)

Theorem (9.39 Lindel ¨of)

Let n ∈N and let E be a subst of Rⁿ. If{Vα}α∈A is a collection of open sets and E ⊆ ∪α∈AVα, then there is an at most countable subset A0 of A such that

E ⊆ [

α∈A0

Vα.

Proof.

LetT be the collection of open balls with rational radii and rational centers, i.e., centers that belong to Qⁿ. This

collection is countable. Moreover, by the proof of the Borel Covering Lemma, T ”approximates” the collection of open balls in the following sense: Given any open ball

Br(x) ⊆Rⁿ, there is a ball B_ρ(a) ∈ T such that x∈B_ρ(a) and Bρ(a) ⊆Br(x).

WEN-CHINGLIEN Advanced Calculus (II)

Theorem (9.39 Lindel ¨of)

Let n ∈N and let E be a subst of Rⁿ. If{Vα}α∈A is a collection of open sets and E ⊆ ∪α∈AVα, then there is an at most countable subset A0 of A such that

E ⊆ [

α∈A0

Vα.

Proof.

LetT be the collection of open balls with rational radii and rational centers, i.e., centers that belong to Qⁿ.This

collection is countable. Moreover, by the proof of the Borel Covering Lemma, T ”approximates” the collection of open balls in the following sense: Given any open ball

Br(x) ⊆Rⁿ, there is a ball B_ρ(a) ∈ T such that x∈B_ρ(a) and Bρ(a) ⊆Br(x).

WEN-CHINGLIEN Advanced Calculus (II)

Theorem (9.39 Lindel ¨of)

Let n ∈N and let E be a subst of Rⁿ. If{Vα}α∈A is a collection of open sets and E ⊆ ∪α∈AVα, then there is an at most countable subset A0 of A such that

E ⊆ [

α∈A0

Vα.

Proof.

LetT be the collection of open balls with rational radii and rational centers, i.e., centers that belong to Qⁿ. This

collection is countable. Moreover, by the proof of the Borel Covering Lemma, T ”approximates” the collection of open balls in the following sense: Given any open ball

Br(x) ⊆Rⁿ, there is a ball B_ρ(a) ∈ T such that x∈B_ρ(a) and Bρ(a) ⊆Br(x).

WEN-CHINGLIEN Advanced Calculus (II)

Theorem (9.39 Lindel ¨of)

Let n ∈N and let E be a subst of Rⁿ. If{Vα}α∈A is a collection of open sets and E ⊆ ∪α∈AVα, then there is an at most countable subset A0 of A such that

E ⊆ [

α∈A0

Vα.

Proof.

LetT be the collection of open balls with rational radii and rational centers, i.e., centers that belong to Qⁿ. This

collection is countable. Moreover, by the proof of the Borel Covering Lemma, T ”approximates” the collection of open balls in the following sense: Given any open ball

Br(x) ⊆Rⁿ,there is a ball B_ρ(a) ∈ T such that x∈B_ρ(a) and Bρ(a) ⊆Br(x).

WEN-CHINGLIEN Advanced Calculus (II)

Theorem (9.39 Lindel ¨of)

Let n ∈N and let E be a subst of Rⁿ. If{Vα}α∈A is a collection of open sets and E ⊆ ∪α∈AVα, then there is an at most countable subset A0 of A such that

E ⊆ [

α∈A0

Vα.

Proof.

LetT be the collection of open balls with rational radii and rational centers, i.e., centers that belong to Qⁿ. This

collection is countable. Moreover, by the proof of the Borel Covering Lemma, T ”approximates” the collection of open balls in the following sense: Given any open ball

Br(x) ⊆Rⁿ, there is a ball B_ρ(a) ∈ T such that x∈B_ρ(a) and Bρ(a) ⊆Br(x).

WEN-CHINGLIEN Advanced Calculus (II)

Theorem (9.39 Lindel ¨of)

Let n ∈N and let E be a subst of Rⁿ. If{Vα}α∈A is a collection of open sets and E ⊆ ∪α∈AVα, then there is an at most countable subset A0 of A such that

E ⊆ [

α∈A0

Vα.

Proof.

LetT be the collection of open balls with rational radii and rational centers, i.e., centers that belong to Qⁿ. This

collection is countable. Moreover, by the proof of the Borel Covering Lemma, T ”approximates” the collection of open balls in the following sense: Given any open ball

Br(x) ⊆Rⁿ,there is a ball B_ρ(a) ∈ T such that x∈B_ρ(a) and Bρ(a) ⊆Br(x).

WEN-CHINGLIEN Advanced Calculus (II)

Theorem (9.39 Lindel ¨of)

Let n ∈N and let E be a subst of Rⁿ. If{Vα}α∈A is a collection of open sets and E ⊆ ∪α∈AVα, then there is an at most countable subset A0 of A such that

E ⊆ [

α∈A0

Vα.

Proof.

LetT be the collection of open balls with rational radii and rational centers, i.e., centers that belong to Qⁿ. This

collection is countable. Moreover, by the proof of the Borel Covering Lemma, T ”approximates” the collection of open balls in the following sense: Given any open ball

Br(x) ⊆Rⁿ, there is a ball B_ρ(a) ∈ T such that x∈B_ρ(a) and Bρ(a) ⊆Br(x).

WEN-CHINGLIEN Advanced Calculus (II)

Proof.

To prove the theorem, let x ∈E.By hypothesis, x∈Vα for someα ∈A. Since Vα is open,there is a r >0 such that Br(x) ⊂Vα. SinceT approximates open balls, we can choose a ball Bx∈ T such that x∈Bx ⊆Vα. The collectionT is countable, hence so is the subcollection

{U1,U2, . . .} := {Bx :x∈E}.

By the choice of the balls Bx, for each k ∈N there is at least oneαk ∈A such that Uk ⊆Vα_k.

Hence, by construction,

E ⊆ [

x∈E

Bx= [

k∈N

Uk ⊆ [

k∈N

Vαk.

Thus, set A0:= {αk :k ∈N}.

WEN-CHINGLIEN Advanced Calculus (II)

Proof.

To prove the theorem, let x ∈E. By hypothesis, x∈Vα for someα ∈A.Since Vα is open, there is a r >0 such that Br(x) ⊂Vα.SinceT approximates open balls, we can choose a ball Bx∈ T such that x∈Bx ⊆Vα. The collectionT is countable, hence so is the subcollection

{U1,U2, . . .} := {Bx :x∈E}.

By the choice of the balls Bx, for each k ∈N there is at least oneαk ∈A such that Uk ⊆Vα_k.

Hence, by construction,

E ⊆ [

x∈E

Bx= [

k∈N

Uk ⊆ [

k∈N

Vαk.

Thus, set A0:= {αk :k ∈N}.

WEN-CHINGLIEN Advanced Calculus (II)

Proof.

To prove the theorem, let x ∈E. By hypothesis, x∈Vα for someα ∈A. Since Vα is open,there is a r >0 such that Br(x) ⊂Vα. SinceT approximates open balls, we can choose a ball Bx∈ T such that x∈Bx ⊆Vα. The collectionT is countable, hence so is the subcollection

{U1,U2, . . .} := {Bx :x∈E}.

By the choice of the balls Bx, for each k ∈N there is at least oneαk ∈A such that Uk ⊆Vα_k.

Hence, by construction,

E ⊆ [

x∈E

Bx= [

k∈N

Uk ⊆ [

k∈N

Vαk.

Thus, set A0:= {αk :k ∈N}.

WEN-CHINGLIEN Advanced Calculus (II)

Proof.

To prove the theorem, let x ∈E. By hypothesis, x∈Vα for someα ∈A. Since Vα is open, there is a r >0 such that Br(x) ⊂Vα.SinceT approximates open balls, we can choose a ball Bx∈ T such that x∈Bx ⊆Vα.The collectionT is countable, hence so is the subcollection

{U1,U2, . . .} := {Bx :x∈E}.

By the choice of the balls Bx, for each k ∈N there is at least oneαk ∈A such that Uk ⊆Vα_k.

Hence, by construction,

E ⊆ [

x∈E

Bx= [

k∈N

Uk ⊆ [

k∈N

Vαk.

Thus, set A0:= {αk :k ∈N}.

WEN-CHINGLIEN Advanced Calculus (II)

Proof.

To prove the theorem, let x ∈E. By hypothesis, x∈Vα for someα ∈A. Since Vα is open, there is a r >0 such that Br(x) ⊂Vα. SinceT approximates open balls, we can choose a ball Bx∈ T such that x∈Bx ⊆Vα. The collectionT is countable, hence so is the subcollection

{U1,U2, . . .} := {Bx :x∈E}.

By the choice of the balls Bx, for each k ∈N there is at least oneαk ∈A such that Uk ⊆Vα_k.

Hence, by construction,

E ⊆ [

x∈E

Bx= [

k∈N

Uk ⊆ [

k∈N

Vαk.

Thus, set A0:= {αk :k ∈N}.

WEN-CHINGLIEN Advanced Calculus (II)

Proof.

To prove the theorem, let x ∈E. By hypothesis, x∈Vα for someα ∈A. Since Vα is open, there is a r >0 such that Br(x) ⊂Vα. SinceT approximates open balls, we can choose a ball Bx∈ T such that x∈Bx ⊆Vα.The collectionT is countable, hence so is the subcollection

{U1,U2, . . .} := {Bx :x∈E}.

By the choice of the balls Bx, for each k ∈N there is at least oneαk ∈A such that Uk ⊆Vα_k.

Hence, by construction,

E ⊆ [

x∈E

Bx= [

k∈N

Uk ⊆ [

k∈N

Vαk.

Thus, set A0:= {αk :k ∈N}.

WEN-CHINGLIEN Advanced Calculus (II)

Proof.

To prove the theorem, let x ∈E. By hypothesis, x∈Vα for someα ∈A. Since Vα is open, there is a r >0 such that Br(x) ⊂Vα. SinceT approximates open balls, we can choose a ball Bx∈ T such that x∈Bx ⊆Vα. The collectionT is countable, hence so is the subcollection

{U1,U2, . . .} := {Bx :x∈E}.

By the choice of the balls Bx, for each k ∈N there is at least oneαk ∈A such that Uk ⊆Vα_k.

Hence, by construction,

E ⊆ [

x∈E

Bx= [

k∈N

Uk ⊆ [

k∈N

Vαk.

Thus, set A0:= {αk :k ∈N}.

WEN-CHINGLIEN Advanced Calculus (II)

Proof.

To prove the theorem, let x ∈E. By hypothesis, x∈Vα for someα ∈A. Since Vα is open, there is a r >0 such that Br(x) ⊂Vα. SinceT approximates open balls, we can choose a ball Bx∈ T such that x∈Bx ⊆Vα. The collectionT is countable, hence so is the subcollection

{U1,U2, . . .} := {Bx :x∈E}.

By the choice of the balls Bx, for each k ∈N there is at least oneαk ∈A such that Uk ⊆Vα_k.

Hence, by construction,

E ⊆ [

x∈E

Bx= [

k∈N

Uk ⊆ [

k∈N

Vαk.

Thus, set A0:= {αk :k ∈N}.

WEN-CHINGLIEN Advanced Calculus (II)

Proof.

To prove the theorem, let x ∈E. By hypothesis, x∈Vα for someα ∈A. Since Vα is open, there is a r >0 such that Br(x) ⊂Vα. SinceT approximates open balls, we can choose a ball Bx∈ T such that x∈Bx ⊆Vα. The collectionT is countable, hence so is the subcollection

{U1,U2, . . .} := {Bx :x∈E}.

By the choice of the balls Bx, for each k ∈N there is at least oneαk ∈A such that Uk ⊆Vα_k.

Hence, by construction,

E ⊆ [

x∈E

Bx= [

k∈N

Uk ⊆ [

k∈N

Vαk.

Thus, set A0:= {αk :k ∈N}.

WEN-CHINGLIEN Advanced Calculus (II)

Proof.

To prove the theorem, let x ∈E. By hypothesis, x∈Vα for someα ∈A. Since Vα is open, there is a r >0 such that Br(x) ⊂Vα. SinceT approximates open balls, we can choose a ball Bx∈ T such that x∈Bx ⊆Vα. The collectionT is countable, hence so is the subcollection

{U1,U2, . . .} := {Bx :x∈E}.

By the choice of the balls Bx, for each k ∈N there is at least oneαk ∈A such that Uk ⊆Vα_k.

Hence, by construction, E ⊆ [

x∈E

Bx= [

k∈N

Uk ⊆ [

k∈N

Vαk.

Thus, set A0:= {αk :k ∈N}.

WEN-CHINGLIEN Advanced Calculus (II)

Proof.

To prove the theorem, let x ∈E. By hypothesis, x∈Vα for someα ∈A. Since Vα is open, there is a r >0 such that Br(x) ⊂Vα. SinceT approximates open balls, we can choose a ball Bx∈ T such that x∈Bx ⊆Vα. The collectionT is countable, hence so is the subcollection

{U1,U2, . . .} := {Bx :x∈E}.

By the choice of the balls Bx, for each k ∈N there is at least oneαk ∈A such that Uk ⊆Vα_k.

Hence, by construction, E ⊆ [

x∈E

Bx= [

k∈N

Uk ⊆ [

k∈N

Vαk.

Thus, set A0:= {αk :k ∈N}.

WEN-CHINGLIEN Advanced Calculus (II)

Thank you.

WEN-CHINGLIEN Advanced Calculus (II)