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Part A Mathematics > Metric Bookwork > Flashcards

Metric Bookwork Flashcards

1
Q

Definition of a metric

A

Positivity + 0 iff 0
Symmetry
Triangle inequality

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2
Q

Definition of norm

A

0 iff 0
||λx|| = λ||x||
||x+y|| <= ||x|| + ||y||

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3
Q

Prove product metric is a metric

A

Page 14

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4
Q

Definition of bounded

A

Contained in an open ball

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5
Q

Prove f continuous iff x->a => f(x)->f(a)

A

Page 18

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6
Q

Prove f continuous iff

{ ||f(x)|| : ||x||<=1 } is bounded

A

Page 18

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7
Q

Definition of isometry

A

Preserves distances

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8
Q

Definition of homeomomorphism

A

Continuous bijection with continuous inverse

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9
Q

Definition of open

A

Y open if for all y in Y there exists a ball around y contained in Y

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10
Q

Union and intersection laws for open and closed sets

A

Any union of open sets is open
A countable intersection of open sets is open

A countable union of closed sets is closed
Any intersection of closed sets is closed

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11
Q

Definition of a neighbourhood

A

N is a neighbourhood of z if there exists an open ball around z in N

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12
Q

Prove f is continuous at a iff the preimage of every neighbourhood of f(a) is a neighbourhood of a

A

Page 29

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13
Q

Prove f is continuous iff the preimage of every open set is open

A

Page 30

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14
Q

Prove if Y

A

Page 32

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15
Q

Define the interior of Y

A

Union of all open sets of X contained in Y

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16
Q

Define the closure of Y

A

Intersection of all closed sets of X that contain Y

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17
Q

Prove a in closure of S iff every open ball of a contains a point of S

A

Page 36

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18
Q

Prove a in closure of S iff there exists a sequence of S whose limit is A
In particular S is closd iff every convergent sequence of S has its limit in S

A

Page 36

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19
Q

Definition of a limit point

A

x is a limit point of S if All balls of x have a point in S other than x

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20
Q

S

A

Page 37

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21
Q

S closure = S U L(S)

A

Page 37

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22
Q

Prove convergent sequences are cauchy, and cauchy sequences are bounded.
Provide counterexamples for the reverse implications

A

Page 39

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23
Q

X complete

Prove Y

A

page 40

24
Q

Let
X complete
and S1 contains S2 contains … are a nested sequence of non-empty closed sets, and diam(Sn)->0

Prove the intersection of Sn contains a unique point

A

Page 41

25
Q

Prove B(X) complete

A

Page 41

26
Q

Prove Cb(X) complete

A

Page 42

27
Q

Define a set to be connected

A

X is connected if

Whenever X=AUB and A,B disjoint and open then one of A or B is empty

28
Q

Prove the following are equivalent
X is connected
A continuous function to {0,1} is constant
The only open and closed subsets of X are X and empty set

A

Page 47

29
Q

Prove Y

A

Page 48

30
Q

State and prove the sunflower lemma

A

Page 48

31
Q

Let A

A

page 49

32
Q

Prove the image of a connected space over a continuous function is connected

A

page 49

33
Q

Define the connected component of x

A

The maximal connected subset of X containing x

34
Q

Prove the connected components of X partition X

A

Page 49

35
Q

Define X path connected

A

If for all a,b in X there is a continuous map {0,1}->X with p(0)=a, p(1)=b

36
Q

Define a path component of x

A

An equivalence class under the equivalence relation a~b if there exists a path from a to b

37
Q

Prove the path components partition the space

Equivalently prove a~b if there exists a path from a to b is an equivalence relation

A

page 52

38
Q

Prove Path connected => connected

A

Page 52

39
Q

Prove a connected open subset of a normed space is path connected

A

Page 52

40
Q

Prove a sequentially compact subspace must be closed and bounded

A

Page 56

41
Q

Prove if X sequentially compact and Y

A

Page 56

42
Q

Prove the image of a sequentially compact space under a continuous map is sequentially compact

Thus a continuous function on a sequentially compact space to R is bounded and attains its boundes

A

Page 57

43
Q

Prove a continuous function from sequentially compact X to R must be uniformly continuous

A

Page 57

44
Q

Prove a sequence in XxY converges iff each part of the sequence converge in X and Y

A

Page 58

45
Q

Prove the product of two sequentially compact metric spaces is sequentially compact

A

Page 58

46
Q

State and prove the Bolzano-Weierstrass theorem

A

Any closed and bounded subset of R^n is sequentially compact

Page 59

47
Q

Prove sequentially compact => complete and bounded

Give a counter example for the reverse implication

A

Page 59

48
Q

What does it mean for X to be totally bounded

A

for all r, X can be covered by finitely many open balls of radius r

49
Q

Prove a metric space is compact iff it is complete and toally bounded

A

Page 60

50
Q

State the Arzela-Ascoli theorem

A

Uniformly bounded
We say F subset C(X) is uniformly bounded if there is an M st ¦f(x)¦<=M for all x in X and for all f in F

Equicontinuous
Let F subset C(X) we say F is equicontinuous if in the definition of continuity delta can be chosen independently of f in F

Arzela-Ascoli
Let X be sequentially compact. Let F subset C(X) be equicontinuous and uniformly bounded. Then any sequence of elements in F has a convergent subsequence. In particular if F is closed then it is sequentially compact

51
Q

Define compactness

A

Every open cover has a finite subcover

52
Q

Define compactness for a subspace Y

A

An open cover of Y is a collection of sets that are open in X that cover Y (not necessarily equal to)
Then every open cover has a finite subcover

53
Q

Let X be compact
Supposed we have a nested sequence S1 in S2 in … of nonempty closed subsets of X
Prove the intersection of Sn is empty

A

Page 66

54
Q

Prove a compact metric space is sequentially compact

A

Page 66

55
Q

State and prove the Heine-Borel theorem

A

The interval [a,b] is compact

Page 67

Part A Mathematics (7 decks)

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