Download Lecture 2: Review of Metric Spaces

Lecture 2: Review of Metric Spaces
Hart Smith
Department of Mathematics
University of Washington, Seattle
Math 524, Autumn 2013
Hart Smith
Math 524
Definition of a Metric Space
A metric space consists of:
a set X , and function (metric) ρ : X × X → [0, ∞) , such that:
ρ(x, y ) = ρ(y , x)
(Symmetry)
ρ(x, y ) = 0 iff x = y
(Non-degeneracy)
ρ(x, y ) + ρ(y , z) ≤ ρ(x, z)
(Triangle inequality)
Examples of metrics on Rn :
Euclidean metric:
ρ(x, y ) =
X
n
j=1
Box metric:
ρ(x, y ) = max |xj − yj |
j
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Math 524
|xj − yj |
2
1/2
Open sets in a metric space
A subset O ⊂ X is open if:
for each x ∈ O there exists δ > 0 ( δ can depend on x )
such that y ∈ O whenever ρ(x, y ) < δ .
B(z, r ) ≡
x : ρ(x, z) < r } is open, by triangle inequality.
The sets X and ∅ are both open.
The union of any collection of open sets is open.
The intersection of a finite collection of open sets is open.
The collection of open subsets of X is a topology on X .
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Math 524
Closed sets in a metric space
A subset F ⊂ X is closed if:
the complement F c ≡ X \F is open.
The intersection of any collection of closed sets is closed.
The union of a finite collection of closed sets is closed.
For any set E ⊂ X , define the interior and the closure of E:
Interior = largest open set contained in E:
E o = int(E) = ∪ O : O ⊂ E
is open
Closure = smallest closed set containing E:
E = ∩F : F ⊃ E
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is closed
Math 524
Sequences in a metric space
A sequence {xn }∞
n=1 ⊂ X converges to x if lim ρ(xn , x) = 0 .
n→∞
A sequence {xn }∞
n=1 ⊂ X is Cauchy if
lim
ρ(xm , xn ) = 0 .
∀ > 0 , ∃ N < ∞ , such that ρ(xm , xn ) < if m, n > N .
m,n→∞
Every convergent sequence is Cauchy
The point x is a cluster point of the sequence {xn }∞
n=1 if, for
every r > 0, B(x, r ) contains xn for infinitely many n.
If x is a cluster point of {xn }∞
n=1 then some sub-sequence
of {xn }∞
converges
to
x.
n=1
If a Cauchy sequence has a cluster point x, then the
sequence converges to x.
Hart Smith
Math 524
Complete metric spaces
A metric space (X , ρ) is complete if:
each Cauchy sequence {xn }∞
n=1 ⊂ X converges to some x ∈ X .
A closed subset E ⊂ X of a complete metric space is
complete; i.e. every Cauchy sequence contained in E
converges to a point in E.
R with Euclidean distance is complete: let x = lim inf xn
Rn with Euclidean distance is complete: let x|j = lim inf xn |j
R\{0} with Euclidean distance is not complete: xn =
A complete metric space has no holes in it.
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Math 524
1
n
Compact sets in a metric space
For E ⊂ X , an open cover of E is:
a collection Oα α∈A of open subsets such that E ⊂ ∪α∈A Oα .
A subset E ⊂ X is compact if:
every open cover Oα α∈A of E has some finite sub-collection
N
Oj j=1 that covers F .
∞
Open interval (0, 1) ⊂ R is not compact: On = ( n1 , 1) n=1 .
A compact set is closed: suppose
E compact and x ∈
/ E.
Let On = y : ρ(y , x) > n1 . Finite cover n ≤ N means
ρ(y , x) ≥ N1 for all y ∈ E, so x ∈
/E, ⇒ E =E.
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Math 524
Sequentially compact sets
A subset E ⊂ X is sequentially compact if:
every sequence {xn }∞
n=1 ⊂ E has a cluster point in E.
E sequentially compact ⇒ E complete
if a Cauchy sequence has a cluster point then it converges.
Theorem
If E ⊂ X is compact then E is sequentially compact.
Proof.
By contradiction: if no point in E is a cluster point of {xn }∞
n=1
then each x has some rx > 0 so that B(x, rx ) contains xn for at
most finitely many n. Some finite number cover E, ⇒⇐
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Math 524
Totally bounded sets
A subset E ⊂ X is totally bounded if:
for every r > 0, E is covered by a finite collection of r -balls:
N
E ⊂ ∪N
n=1 B(xn , r ) for some finite collection {xn }n=1 ⊂ E.
A compact set E is totally bounded.
For subsets E ⊂ Rn , a set is totally bounded if and only if it
is contained in B(0, R) for some R < ∞.
We have shown:
A compact set E is sequentially compact.
A compact set E is complete, and it is totally bounded.
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Math 524
Theorem
For subsets E of a metric space, the following are equivalent:
(i) E is compact.
(ii) E is sequentially compact.
(iii) E is complete and totally bounded.
Scheme of proof:
Have shown (i) ⇒ (ii) and (i) ⇒ (iii).
Will show (iii) ⇔ (ii), and then (iii) & (ii) ⇒ (i).
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Math 524
Complete & totally bounded ⇒ sequentially compact
Given sequence {xn }∞
n=1 ⊂ E, need construct cluster point x.
Idea: if yj → x, and B(yj , 2−j ) contains infinitely many elements
of {xn }, then x is a cluster point: B(x, r ) ⊃ B(yj , 2−j ) if j large.
Take finite cover of E by 2−1 balls:
∃ y1 ∈ E : B(y1 , 2−1 ) contains infinitely many elements of {xn }.
Take finite cover of B(y1 , 2−1 ) ∩ E by 2−2 balls:
∃ y2 ∈ B(y1 , 2−1 ) ∩ E : B(y2 , 2−2 ) contains infinitely many {xn }.
∃ yj+1 ∈ B(yj , 2−j ) : B(yj+1 , 2−j−1 ) ⊃ infinitely many {xn }.
The sequence {yj } is Cauchy since ρ(yj+1 , yj ) ≤ 2−j .
By completeness of E, yj → x for some x ∈ E.
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Math 524
Sequentially compact ⇒ complete & totally bounded
Sequentially compact ⇒ complete is easy:
If Cauchy {xn } ⊂ E has a cluster point x ∈ E, it converges to x.
Not totally bounded ⇒ not sequentially compact:
If not totally bounded, ∃ r > 0 : E 6⊂ ∪nj=1 B(xj , r ) for any {xj }.
Choose x1 ∈ E
Choose x2 ∈ E : x2 ∈
/ B(x1 , r )
···
Choose xn ∈ E : xn+1 ∈
/ ∪nj=1 B(xj , r )
Result: a sequence {xn }∞
n=1 such that ρ(xm , xn ) > r for all m, n,
so it cannot have a cluster point .
Hart Smith
Math 524
Sequentially compact & totally bounded ⇒ compact
Claim: if E is sequentially compact, and E ⊂ ∪α Oα
then there exists r > 0 : ∀x ∈ E , B(x, r ) ⊂ Oα for some α.
Suppose not: take xn ∈ E such that B(xn , 2−n ) 6⊂ Oα for any α.
The sequence {xn }∞
n=1 has a cluster point x ∈ E.
For some α , x ∈ Oα , so B(x, r ) ⊂ Oα some r > 0.
r
r
, and ρ(xn , x) < .
2
2
Then B(xn , 2−n ) ⊂ B(x, r ) ⊂ Oα , ⇒⇐
Take n such that 2−n <
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Math 524
Remarks
Compactness of E ⊂ X is a topological property:
it depends only on the collection of open subsets of X .
Let (X1 , ρ1 ) and (X2 , ρ2 ) be metric spaces
If F : X1 ↔ X2 is a 1-1, onto mapping of sets, and both
F and F −1 map opens sets to open sets, then F and F −1
map compact sets to compact sets.
Total boundedness & completeness depend on the metric:
Consider the map:
1-1, onto
tan x : (− π2 , π2 ) : ←→ R
open intervals ↔ open intervals ⇒ open sets ↔ open sets.
(− π2 , π2 ) is totally bounded, not complete (Euclidean metric)
R is complete, but not totally bounded.
Hart Smith
Math 524
Equivalence of metrics
Definition
Two metrics ρ1 and ρ2 on X are equivalent if there is C > 0:
ρ1 (x, y ) ≤ C ρ2 (x, y ) ,
ρ2 (x, y ) ≤ C ρ1 (x, y ) .
All basic metric space notions are equivalent for ρ1 for ρ2 :
Cauchy sequence, completeness, total boundedness, . . .
The metrics ρ1 (x, y ) = | tan x − tan y | , ρ2 (x, y ) = |x − y |
on (− π2 , π2 ) are not equivalent, but give the same collection
of open sets.
Hart Smith
Math 524