Metric Bookwork

Question 1

Definition of a metric

Answer 1

Positivity + 0 iff 0
Symmetry
Triangle inequality

Question 2

Definition of norm

Answer 2

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

Question 3

Prove product metric is a metric

Answer 3

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Question 4

Definition of bounded

Answer 4

Contained in an open ball

Question 5

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

Answer 5

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Question 6

Prove f continuous iff

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

Answer 6

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Question 7

Definition of isometry

Answer 7

Preserves distances

Question 8

Definition of homeomomorphism

Answer 8

Continuous bijection with continuous inverse

Question 9

Definition of open

Answer 9

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

Question 10

Union and intersection laws for open and closed sets

Answer 10

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

Question 11

Definition of a neighbourhood

Answer 11

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

Question 12

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

Answer 12

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Question 13

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

Answer 13

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Question 14

Prove if Y

Answer 14

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Question 15

Define the interior of Y

Answer 15

Union of all open sets of X contained in Y

Question 16

Define the closure of Y

Answer 16

Intersection of all closed sets of X that contain Y

Question 17

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

Answer 17

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Question 18

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

Answer 18

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Question 19

Definition of a limit point

Answer 19

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

Question 20

S

Answer 20

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Question 21

S closure = S U L(S)

Answer 21

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Question 22

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

Answer 22

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Question 23

X complete

Prove Y

Answer 23

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Question 24

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

Answer 24

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Question 25

Prove B(X) complete

Answer 25

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Question 26

Prove Cb(X) complete

Answer 26

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Question 27

Define a set to be connected

Answer 27

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

Question 28

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

Answer 28

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Question 29

Prove Y

Answer 29

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Question 30

State and prove the sunflower lemma

Answer 30

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Question 31

Let A

Answer 31

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Question 32

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

Answer 32

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Question 33

Define the connected component of x

Answer 33

The maximal connected subset of X containing x

Question 34

Prove the connected components of X partition X

Answer 34

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Question 35

Define X path connected

Answer 35

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

Question 36

Define a path component of x

Answer 36

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

Question 37

Prove the path components partition the space | Equivalently prove a~b if there exists a path from a to b is an equivalence relation

Answer 37

page 52

Question 38

Prove Path connected => connected

Answer 38

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Question 39

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

Answer 39

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Question 40

Prove a sequentially compact subspace must be closed and bounded

Answer 40

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Question 41

Prove if X sequentially compact and Y

Answer 41

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Question 42

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

Answer 42

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Question 43

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

Answer 43

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Question 44

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

Answer 44

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Question 45

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

Answer 45

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Question 46

State and prove the Bolzano-Weierstrass theorem

Answer 46

Any closed and bounded subset of R^n is sequentially compact | Page 59

Question 47

Prove sequentially compact => complete and bounded | Give a counter example for the reverse implication

Answer 47

Page 59

Question 48

What does it mean for X to be totally bounded

Answer 48

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

Question 49

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

Answer 49

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Question 50

State the Arzela-Ascoli theorem

Answer 50

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

Question 51

Define compactness

Answer 51

Every open cover has a finite subcover

Question 52

Define compactness for a subspace Y

Answer 52

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

Question 53

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

Answer 53

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Question 54

Prove a compact metric space is sequentially compact

Answer 54

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Question 55

State and prove the Heine-Borel theorem

Answer 55

The interval [a,b] is compact | Page 67