## 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*

^{n}### 9.4: Compact Sets

Definition (9.35)

LetV = {Vα}α∈A **be a collection of subsets of R*** ^{n}*, and

*suppose that E*⊆

**R**

*.*

^{n}(i)V *is said to cover E (or be a covering of E) if and only if*
*E* ⊆ [

α∈A

*V*α.

(ii)V *is said to be an open covering of E if and only if*V
*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 A*0 *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 R**^{n}*. If*{Vα}α∈A *is a*
*collection of open sets and E* ⊆ ∪α∈A*V*α*, then there is an*
*at most countable subset A*0 *of A such that*

*E* ⊆ [

α∈A0

*V*α.

Proof.

LetT be the collection of open balls with rational radii and
**rational centers, i.e., centers that belong to Q*** ^{n}*.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

*B**r*(x) ⊆**R**^{n}*, there is a ball B*_{ρ}(a) ∈ T **such that x**∈*B*_{ρ}(a)
*and B*ρ(a) ⊆*B**r*(x).

WEN-CHINGLIEN **Advanced Calculus (II)**

Theorem (9.39 Lindel ¨of)

*Let n* ∈**N and let E be a subst of R**^{n}*. If*{Vα}α∈A *is a*
*collection of open sets and E* ⊆ ∪α∈A*V*α*, then there is an*
*at most countable subset A*0 *of A such that*

*E* ⊆ [

α∈A0

*V*α.

Proof.

LetT be the collection of open balls with rational radii and
rational centers, **i.e., centers that belong to Q*** ^{n}*. 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

*B**r*(x) ⊆**R**^{n}*, there is a ball B*_{ρ}(a) ∈ T **such that x**∈*B*_{ρ}(a)
*and B*ρ(a) ⊆*B**r*(x).

WEN-CHINGLIEN **Advanced Calculus (II)**

Theorem (9.39 Lindel ¨of)

*Let n* ∈**N and let E be a subst of R**^{n}*. If*{Vα}α∈A *is a*
*collection of open sets and E* ⊆ ∪α∈A*V*α*, then there is an*
*at most countable subset A*0 *of A such that*

*E* ⊆ [

α∈A0

*V*α.

Proof.

LetT be the collection of open balls with rational radii and
**rational centers, i.e., centers that belong to Q*** ^{n}*.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

*B**r*(x) ⊆**R**^{n}*, there is a ball B*_{ρ}(a) ∈ T **such that x**∈*B*_{ρ}(a)
*and B*ρ(a) ⊆*B**r*(x).

WEN-CHINGLIEN **Advanced Calculus (II)**

Theorem (9.39 Lindel ¨of)

*Let n* ∈**N and let E be a subst of R**^{n}*. If*{Vα}α∈A *is a*
*collection of open sets and E* ⊆ ∪α∈A*V*α*, then there is an*
*at most countable subset A*0 *of A such that*

*E* ⊆ [

α∈A0

*V*α.

Proof.

LetT be the collection of open balls with rational radii and
**rational centers, i.e., centers that belong to Q*** ^{n}*. This

*B**r*(x) ⊆**R**^{n}*, there is a ball B*_{ρ}(a) ∈ T **such that x**∈*B*_{ρ}(a)
*and B*ρ(a) ⊆*B**r*(x).

WEN-CHINGLIEN **Advanced Calculus (II)**

Theorem (9.39 Lindel ¨of)

*Let n* ∈**N and let E be a subst of R**^{n}*. If*{Vα}α∈A *is a*
*collection of open sets and E* ⊆ ∪α∈A*V*α*, then there is an*
*at most countable subset A*0 *of A such that*

*E* ⊆ [

α∈A0

*V*α.

Proof.

LetT be the collection of open balls with rational radii and
**rational centers, i.e., centers that belong to Q*** ^{n}*. This

*B**r*(x) ⊆**R*** ^{n}*,

*there is a ball B*

_{ρ}(a) ∈ T

**such that x**∈

*B*

_{ρ}(a)

*and B*ρ(a) ⊆

*B*

*r*(x).

WEN-CHINGLIEN **Advanced Calculus (II)**

Theorem (9.39 Lindel ¨of)

*Let n* ∈**N and let E be a subst of R**^{n}*. If*{Vα}α∈A *is a*
*collection of open sets and E* ⊆ ∪α∈A*V*α*, then there is an*
*at most countable subset A*0 *of A such that*

*E* ⊆ [

α∈A0

*V*α.

Proof.

**rational centers, i.e., centers that belong to Q*** ^{n}*. This

*B**r*(x) ⊆**R**^{n}*, there is a ball B*_{ρ}(a) ∈ T **such that x**∈*B*_{ρ}(a)
*and B*ρ(a) ⊆*B**r*(x).

WEN-CHINGLIEN **Advanced Calculus (II)**

Theorem (9.39 Lindel ¨of)

*Let n* ∈**N and let E be a subst of R**^{n}*. If*{Vα}α∈A *is a*
*collection of open sets and E* ⊆ ∪α∈A*V*α*, then there is an*
*at most countable subset A*0 *of A such that*

*E* ⊆ [

α∈A0

*V*α.

Proof.

**rational centers, i.e., centers that belong to Q*** ^{n}*. This

*B**r*(x) ⊆**R*** ^{n}*,

*there is a ball B*

_{ρ}(a) ∈ T

**such that x**∈

*B*

_{ρ}(a)

*and B*ρ(a) ⊆

*B*

*r*(x).

WEN-CHINGLIEN **Advanced Calculus (II)**

Theorem (9.39 Lindel ¨of)

*Let n* ∈**N and let E be a subst of R**^{n}*. If*{Vα}α∈A *is a*
*collection of open sets and E* ⊆ ∪α∈A*V*α*, then there is an*
*at most countable subset A*0 *of A such that*

*E* ⊆ [

α∈A0

*V*α.

Proof.

**rational centers, i.e., centers that belong to Q*** ^{n}*. This

*B**r*(x) ⊆**R**^{n}*, there is a ball B*_{ρ}(a) ∈ T **such that x**∈*B*_{ρ}(a)
*and B*ρ(a) ⊆*B**r*(x).

WEN-CHINGLIEN **Advanced Calculus (II)**

Proof.

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

{U1,*U*2, . . .} := {B**x** :**x**∈*E}.*

*By the choice of the balls B***x***, for each k* ∈**N there is at**
least oneα*k* ∈*A such that U**k* ⊆*V*α* _{k}*.

Hence, by construction,

*E* ⊆ [

**x**∈*E*

*B***x**= [

*k*∈**N**

*U**k* ⊆ [

*k*∈**N**

*V*αk.

*Thus, set A*0:= {α*k* :*k* ∈**N}.**

WEN-CHINGLIEN **Advanced Calculus (II)**

Proof.

**To prove the theorem, let x** ∈* E. By hypothesis, x*∈

*V*α for someα ∈

*A.Since V*α

*is open, there is a r*>0 such that

*B*

*r*(x) ⊂

**V**α.SinceT approximates open balls, we can

*choose a ball B*

**x**∈ T

**such that x**∈

*B*

**x**⊆

*V*α. The collectionT is countable, hence so is the subcollection

{U1,*U*2, . . .} := {B**x** :**x**∈*E}.*

*By the choice of the balls B***x***, for each k* ∈**N there is at**
least oneα*k* ∈*A such that U**k* ⊆*V*α* _{k}*.

Hence, by construction,

*E* ⊆ [

**x**∈*E*

*B***x**= [

*k*∈**N**

*U**k* ⊆ [

*k*∈**N**

*V*αk.

*Thus, set A*0:= {α*k* :*k* ∈**N}.**

WEN-CHINGLIEN **Advanced Calculus (II)**

Proof.

**To prove the theorem, let x** ∈* E. By hypothesis, x*∈

*V*α for someα ∈

*A. Since V*α is open,

*there is a r*>0 such that

*B*

*r*(x) ⊂

**V**α. SinceT approximates open balls, we can

*choose a ball B*

**x**∈ T

**such that x**∈

*B*

**x**⊆

*V*α. The collectionT is countable, hence so is the subcollection

{U1,*U*2, . . .} := {B**x** :**x**∈*E}.*

*By the choice of the balls B***x***, for each k* ∈**N there is at**
least oneα*k* ∈*A such that U**k* ⊆*V*α* _{k}*.

Hence, by construction,

*E* ⊆ [

**x**∈*E*

*B***x**= [

*k*∈**N**

*U**k* ⊆ [

*k*∈**N**

*V*αk.

*Thus, set A*0:= {α*k* :*k* ∈**N}.**

WEN-CHINGLIEN **Advanced Calculus (II)**

Proof.

**To prove the theorem, let x** ∈* E. By hypothesis, x*∈

*V*α for someα ∈

*A. Since V*α

*is open, there is a r*>0 such that

*B*

*r*(x) ⊂

**V**α.SinceT approximates open balls, we can

*choose a ball B*

**x**∈ T

**such that x**∈

*B*

**x**⊆

*V*α.The collectionT is countable, hence so is the subcollection

{U1,*U*2, . . .} := {B**x** :**x**∈*E}.*

*By the choice of the balls B***x***, for each k* ∈**N there is at**
least oneα*k* ∈*A such that U**k* ⊆*V*α* _{k}*.

Hence, by construction,

*E* ⊆ [

**x**∈*E*

*B***x**= [

*k*∈**N**

*U**k* ⊆ [

*k*∈**N**

*V*αk.

*Thus, set A*0:= {α*k* :*k* ∈**N}.**

WEN-CHINGLIEN **Advanced Calculus (II)**

Proof.

**To prove the theorem, let x** ∈* E. By hypothesis, x*∈

*V*α for someα ∈

*A. Since V*α

*is open, there is a r*>0 such that

*B*

*r*(x) ⊂

**V**α. SinceT approximates open balls, we can

*choose a ball B*

**x**∈ T

**such that x**∈

*B*

**x**⊆

*V*α. The collectionT is countable, hence so is the subcollection

{U1,*U*2, . . .} := {B**x** :**x**∈*E}.*

*By the choice of the balls B***x***, for each k* ∈**N there is at**
least oneα*k* ∈*A such that U**k* ⊆*V*α* _{k}*.

Hence, by construction,

*E* ⊆ [

**x**∈*E*

*B***x**= [

*k*∈**N**

*U**k* ⊆ [

*k*∈**N**

*V*αk.

*Thus, set A*0:= {α*k* :*k* ∈**N}.**

WEN-CHINGLIEN **Advanced Calculus (II)**

Proof.

**To prove the theorem, let x** ∈* E. By hypothesis, x*∈

*V*α for someα ∈

*A. Since V*α

*is open, there is a r*>0 such that

*B*

*r*(x) ⊂

**V**α. SinceT approximates open balls, we can

*choose a ball B*

**x**∈ T

**such that x**∈

*B*

**x**⊆

*V*α.The collectionT is countable, hence so is the subcollection

{U1,*U*2, . . .} := {B**x** :**x**∈*E}.*

*By the choice of the balls B***x***, for each k* ∈**N there is at**
least oneα*k* ∈*A such that U**k* ⊆*V*α* _{k}*.

Hence, by construction,

*E* ⊆ [

**x**∈*E*

*B***x**= [

*k*∈**N**

*U**k* ⊆ [

*k*∈**N**

*V*αk.

*Thus, set A*0:= {α*k* :*k* ∈**N}.**

WEN-CHINGLIEN **Advanced Calculus (II)**

Proof.

**To prove the theorem, let x** ∈* E. By hypothesis, x*∈

*V*α for someα ∈

*A. Since V*α

*is open, there is a r*>0 such that

*B*

*r*(x) ⊂

**V**α. SinceT approximates open balls, we can

*choose a ball B*

**x**∈ T

**such that x**∈

*B*

**x**⊆

*V*α. The collectionT is countable, hence so is the subcollection

{U1,*U*2, . . .} := {B**x** :**x**∈*E}.*

*By the choice of the balls B***x***, for each k* ∈**N there is at**
least oneα*k* ∈*A such that U**k* ⊆*V*α* _{k}*.

Hence, by construction,

*E* ⊆ [

**x**∈*E*

*B***x**= [

*k*∈**N**

*U**k* ⊆ [

*k*∈**N**

*V*αk.

*Thus, set A*0:= {α*k* :*k* ∈**N}.**

WEN-CHINGLIEN **Advanced Calculus (II)**

Proof.

**To prove the theorem, let x** ∈* E. By hypothesis, x*∈

*V*α for someα ∈

*A. Since V*α

*is open, there is a r*>0 such that

*B*

*r*(x) ⊂

**V**α. SinceT approximates open balls, we can

*choose a ball B*

**x**∈ T

**such that x**∈

*B*

**x**⊆

*V*α. The collectionT is countable, hence so is the subcollection

{U1,*U*2, . . .} := {B**x** :**x**∈*E}.*

*By the choice of the balls B***x***, for each k* ∈**N there is at**
least oneα*k* ∈*A such that U**k* ⊆*V*α* _{k}*.

Hence, by construction,

*E* ⊆ [

**x**∈*E*

*B***x**= [

*k*∈**N**

*U**k* ⊆ [

*k*∈**N**

*V*αk.

*Thus, set A*0:= {α*k* :*k* ∈**N}.**

WEN-CHINGLIEN **Advanced Calculus (II)**

Proof.

**To prove the theorem, let x** ∈* E. By hypothesis, x*∈

*V*α for someα ∈

*A. Since V*α

*is open, there is a r*>0 such that

*B*

*r*(x) ⊂

**V**α. SinceT approximates open balls, we can

*choose a ball B*

**x**∈ T

**such that x**∈

*B*

**x**⊆

*V*α. The collectionT is countable, hence so is the subcollection

{U1,*U*2, . . .} := {B**x** :**x**∈*E}.*

*By the choice of the balls B***x***, for each k* ∈**N there is at**
least oneα*k* ∈*A such that U**k* ⊆*V*α* _{k}*.

Hence, by construction,

*E* ⊆ [

**x**∈*E*

*B***x**= [

*k*∈**N**

*U**k* ⊆ [

*k*∈**N**

*V*αk.

*Thus, set A*0:= {α*k* :*k* ∈**N}.**

WEN-CHINGLIEN **Advanced Calculus (II)**

Proof.

**To prove the theorem, let x** ∈* E. By hypothesis, x*∈

*V*α for someα ∈

*A. Since V*α

*is open, there is a r*>0 such that

*B*

*r*(x) ⊂

**V**α. SinceT approximates open balls, we can

*choose a ball B*

**x**∈ T

**such that x**∈

*B*

**x**⊆

*V*α. The collectionT is countable, hence so is the subcollection

{U1,*U*2, . . .} := {B**x** :**x**∈*E}.*

*By the choice of the balls B***x***, for each k* ∈**N there is at**
least oneα*k* ∈*A such that U**k* ⊆*V*α* _{k}*.

Hence, by construction,
*E* ⊆ [

**x**∈*E*

*B***x**= [

*k*∈**N**

*U**k* ⊆ [

*k*∈**N**

*V*αk.

*Thus, set A*0:= {α*k* :*k* ∈**N}.**

WEN-CHINGLIEN **Advanced Calculus (II)**

Proof.

**To prove the theorem, let x** ∈* E. By hypothesis, x*∈

*V*α for someα ∈

*A. Since V*α

*is open, there is a r*>0 such that

*B*

*r*(x) ⊂

**V**α. SinceT approximates open balls, we can

*choose a ball B*

**x**∈ T

**such that x**∈

*B*

**x**⊆

*V*α. The collectionT is countable, hence so is the subcollection

{U1,*U*2, . . .} := {B**x** :**x**∈*E}.*

*By the choice of the balls B***x***, for each k* ∈**N there is at**
least oneα*k* ∈*A such that U**k* ⊆*V*α* _{k}*.

Hence, by construction,
*E* ⊆ [

**x**∈*E*

*B***x**= [

*k*∈**N**

*U**k* ⊆ [

*k*∈**N**

*V*αk.

*Thus, set A*0:= {α*k* :*k* ∈**N}.**

WEN-CHINGLIEN **Advanced Calculus (II)**

*Thank you.*

WEN-CHINGLIEN **Advanced Calculus (II)**