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MA 471G (Advanced Calculus) > Chapter 3 > Flashcards

Chapter 3 Flashcards

1
Q

A set O c R is open if…

A

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

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2
Q

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

A

A set O c R is open if…

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3
Q

Theorem-

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

A

i. ) … is open
ii. ) is open.

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4
Q

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

A

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

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5
Q

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

A

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

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6
Q

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

A

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

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7
Q

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

A

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

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8
Q

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

A

it is not a limit a point of A.

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9
Q

it is not a limit a point of A.

A

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

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10
Q

A set F c R is closed if…

A

F contains all its limit points.

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11
Q

F contains all its limit points.

A

A set F c R is closed if…

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12
Q

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

A

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

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13
Q

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

A

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

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14
Q

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

A

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

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15
Q

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

A

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

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16
Q

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

The closure of A is defined as…

A

Ā = A U L

17
Q

Ā = A U L

A

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

The closure of A is defined as…

18
Q

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

A

the smallest closed set containing A.

19
Q

the smallest closed set containing A.

A

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

20
Q

Theorem-

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

A

i. ) the complement Oc is closed.
ii. ) the complement Fc is open.

21
Q

i. ) the complement Oc is closed.
ii. ) the complement Fc is open.

A

Theorem-

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

22
Q

A set K c R is compact if…

A

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

23
Q

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

A

A set K c R is compact if…

24
Q

A set A c R is bounded if…

A

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

25
Q

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

A

A set A c R is bounded if…

26
Q

Characterization of Compactness in R- A set K c R is compact…

A

iff it is closed and bounded.

27
Q

iff it is closed and bounded.

A

Characterization of Compactness in R- A set K c R is compact…

28
Q

Nested Compact Set Property-

If … K2 c K1

is a nested sequence of non-empty compact sets, then…

A

the intersection of Kn is not empty.

29
Q

the intersection of Kn is not empty.

A

Nested Compact Set Property-

If … K2 c K1

is a nested sequence of non-empty compact sets, then…

30
Q

Let A c R. An open conver for A is…

A

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

31
Q

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

A

Let A c R. An open conver for A is…

32
Q

Given an open cover for A, a finite subcover…

A

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

33
Q

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

A

Given an open cover for A, a finite subcover…

34
Q

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:

A

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

35
Q

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

A

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:

MA 471G (Advanced Calculus) (9 decks)

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