Metric Bookwork Flashcards
Definition of a metric
Positivity + 0 iff 0
Symmetry
Triangle inequality
Definition of norm
0 iff 0
||λx|| = λ||x||
||x+y|| <= ||x|| + ||y||
Prove product metric is a metric
Page 14
Definition of bounded
Contained in an open ball
Prove f continuous iff x->a => f(x)->f(a)
Page 18
Prove f continuous iff
{ ||f(x)|| : ||x||<=1 } is bounded
Page 18
Definition of isometry
Preserves distances
Definition of homeomomorphism
Continuous bijection with continuous inverse
Definition of open
Y open if for all y in Y there exists a ball around y contained in Y
Union and intersection laws for open and closed sets
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
Definition of a neighbourhood
N is a neighbourhood of z if there exists an open ball around z in N
Prove f is continuous at a iff the preimage of every neighbourhood of f(a) is a neighbourhood of a
Page 29
Prove f is continuous iff the preimage of every open set is open
Page 30
Prove if Y
Page 32
Define the interior of Y
Union of all open sets of X contained in Y
Define the closure of Y
Intersection of all closed sets of X that contain Y
Prove a in closure of S iff every open ball of a contains a point of S
Page 36
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
Page 36
Definition of a limit point
x is a limit point of S if All balls of x have a point in S other than x
S
Page 37
S closure = S U L(S)
Page 37
Prove convergent sequences are cauchy, and cauchy sequences are bounded.
Provide counterexamples for the reverse implications
Page 39
X complete
Prove Y
page 40
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
Page 41
Prove B(X) complete
Page 41
Prove Cb(X) complete
Page 42
Define a set to be connected
X is connected if
Whenever X=AUB and A,B disjoint and open then one of A or B is empty
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
Page 47
Prove Y
Page 48
State and prove the sunflower lemma
Page 48
Let A
page 49
Prove the image of a connected space over a continuous function is connected
page 49
Define the connected component of x
The maximal connected subset of X containing x
Prove the connected components of X partition X
Page 49
Define X path connected
If for all a,b in X there is a continuous map {0,1}->X with p(0)=a, p(1)=b
Define a path component of x
An equivalence class under the equivalence relation a~b if there exists a path from a to b
Prove the path components partition the space
Equivalently prove a~b if there exists a path from a to b is an equivalence relation
page 52
Prove Path connected => connected
Page 52
Prove a connected open subset of a normed space is path connected
Page 52
Prove a sequentially compact subspace must be closed and bounded
Page 56
Prove if X sequentially compact and Y
Page 56
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
Page 57
Prove a continuous function from sequentially compact X to R must be uniformly continuous
Page 57
Prove a sequence in XxY converges iff each part of the sequence converge in X and Y
Page 58
Prove the product of two sequentially compact metric spaces is sequentially compact
Page 58
State and prove the Bolzano-Weierstrass theorem
Any closed and bounded subset of R^n is sequentially compact
Page 59
Prove sequentially compact => complete and bounded
Give a counter example for the reverse implication
Page 59
What does it mean for X to be totally bounded
for all r, X can be covered by finitely many open balls of radius r
Prove a metric space is compact iff it is complete and toally bounded
Page 60
State the Arzela-Ascoli theorem
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
Define compactness
Every open cover has a finite subcover
Define compactness for a subspace Y
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
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
Page 66
Prove a compact metric space is sequentially compact
Page 66
State and prove the Heine-Borel theorem
The interval [a,b] is compact
Page 67
