Chapter 3

Question 1

A set O c R is open if…

Answer 1

for all points a in O, there exists an ε-neighborhood V_ε(a) c O

Question 2

for all points a in O, there exists an ε-neighborhood V_ε(a) c O

Answer 2

A set O c R is open if…

Question 3

Theorem-

i. ) The union of an arbitrary collection of open sets is…
ii. ) The intersection of a finite collection of open sets…

Answer 3

i. ) … is open
ii. ) is open.

Question 4

A point is a limit point of a set A if…

Answer 4

every ε-neighborhood V_ε(x) of x intersects the set A at some point other than x.

Question 5

every ε-neighborhood V_ε(x) of x intersects the set A at some point other than x.

Answer 5

A point is a limit point of a set A if…

Question 6

Theorem- A point x is a limit point of a set iff…

Answer 6

x = lim a_n for some sequence (a_n) contained in A satisfying a_n not equal to x for all n in N.

Question 7

x = lim a_n for some sequence (a_n) contained in A satisfying a_n not equal to x for all n in N.

Answer 7

Theorem- A point x is a limit point of a set iff…

Question 8

A point a in A is an isolated point of A if…

Answer 8

it is not a limit a point of A.

Question 9

it is not a limit a point of A.

Answer 9

A point a in A is an isolated point of A if…

Question 10

A set F c R is closed if…

Answer 10

F contains all its limit points.

Question 11

F contains all its limit points.

Answer 11

A set F c R is closed if…

Question 12

Theorem- A set F c R is closed if and only if…

Answer 12

every Cauchy sequence contained in F has a limit that is also an element of F.

Question 13

every Cauchy sequence contained in F has a limit that is also an element of F.

Answer 13

Theorem- A set F c R is closed if and only if…

Question 14

Density of Q in R- for every y in R, …

Answer 14

there exists a sequence of rational numbers that converges to y.

Question 15

there exists a sequence of rational numbers that converges to y.

Answer 15

Density of Q in R- for every y in R, …

Question 16

Given a set A c R, let L be the set of all limit points of A.

The closure of A is defined as…

Answer 16

Ā = A U L

Question 17

Ā = A U L

Answer 17

Given a set A c R, let L be the set of all limit points of A.

The closure of A is defined as…

Question 18

Theorem- For any A c R, the closure Ā is a closed set and …

Answer 18

the smallest closed set containing A.

Question 19

the smallest closed set containing A.

Answer 19

Theorem- For any A c R, the closure Ā is a closed set and …

Question 20

Theorem-

i. ) A set O is open iff…
ii. ) A set F is closed iff…

Answer 20

i. ) the complement O^c is closed.
ii. ) the complement F^c is open.

Question 21

i. ) the complement O^c is closed.
ii. ) the complement F^c is open.

Answer 21

Theorem-

i. ) A set O is open iff…
ii. ) A set F is closed iff…

Question 22

A set K c R is compact if…

Answer 22

every sequence in K has a subsequence that converges to a limit that is also in K.

Question 23

every sequence in K has a subsequence that converges to a limit that is also in K.

Answer 23

A set K c R is compact if…

Question 24

A set A c R is bounded if…

Answer 24

there exists an M > 0 such that |a| < M for all a in A.

Question 25

there exists an M \> 0 such that |a| _\<_ M for all a in A.

Answer 25

A set A _c_ **R** is bounded if...

Question 26

Characterization of Compactness in **R**- A set K _c_ **R** is compact...

Answer 26

iff it is closed and bounded.

Question 27

iff it is closed and bounded.

Answer 27

Characterization of Compactness in **R**- A set K _c_ **R** is compact...

Question 28

Nested Compact Set Property- If ... K₂ _c_ K₁ is a nested sequence of non-empty compact sets, then...

Answer 28

the intersection of K_n is not empty.

Question 29

the intersection of K_n is not empty.

Answer 29

Nested Compact Set Property- If ... K₂ _c_ K₁ is a nested sequence of non-empty compact sets, then...

Question 30

Let A _c_ **R**. An open conver for A is...

Answer 30

a possibly infinite collection of open sets whose union contains the set A.

Question 31

a possibly infinite collection of open sets whose union contains the set A.

Answer 31

Let A _c_ **R**. An open conver for A is...

Question 32

Given an open cover for A, a finite subcover...

Answer 32

is a finite subcollection of open sets from the original open cover whose union still contains A.

Question 33

is a finite subcollection of open sets from the original open cover whose union still contains A.

Answer 33

Given an open cover for A, a finite subcover...

Question 34

Heine-Borel Theorem- Let K be a subset of **R**. All of the following statements are equivalent in the sense that any one of them implies the two others:

Answer 34

i. ) K is compact. ii. ) K is closed and bounded. iii. ) Every open cover for K has a finite subcover.

Question 35

i. ) K is compact. ii. ) K is closed and bounded. iii. ) Every open cover for K has a finite subcover.

Answer 35

Heine-Borel Theorem- Let K be a subset of **R**. All of the following statements are equivalent in the sense that any one of them implies the two others: