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: n 1/2 X 2 Euclidean metric: ρ(x, y) = |xj − yj | j=1
Box metric: ρ(x, y) = max |xj − yj | j
Hart Smith Math 524 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.
Hart Smith 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:
Eo = int(E) = ∪ O : O ⊂ E is open
Closure = smallest closed set containing E:
E = ∩ F : F ⊃ E is closed
Hart Smith Math 524 Sequences in a metric space
∞ A sequence {xn} ⊂ X converges to x if lim ρ(xn, x) = 0 . n=1 n→∞
∞ A sequence {xn} ⊂ X is Cauchy if lim ρ(xm, xn) = 0 . n=1 m,n→∞ ∀ > 0 , ∃ N < ∞ , such that ρ(xm, xn) < if m, n > 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}n=1 converges to x. 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
n R with Euclidean distance is complete: let x|j = lim inf xn|j
1 R\{0} with Euclidean distance is not complete: xn = n
A complete metric space has no holes in it.
Hart Smith Math 524 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∈AOα .
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.
1 ∞ Open interval (0, 1) ⊂ R is not compact: On = ( n , 1) n=1 . A compact set is closed: suppose E compact and x ∈/ E. 1 Let On = y : ρ(y, x) > n . Finite cover n ≤ N means 1 ρ(y, x) ≥ N for all y ∈ E, so x ∈/ E , ⇒ E = E .
Hart Smith 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, ⇒⇐
Hart Smith 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 N E ⊂ ∪n=1B(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.
Hart Smith 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).
Hart Smith Math 524 Complete & totally bounded ⇒ sequentially compact
∞ Given sequence {xn}n=1 ⊂ E, need construct cluster point x.
−j Idea: if yj → x, and B(yj , 2 ) contains infinitely many elements −j of {xn}, then x is a cluster point: B(x, r) ⊃ B(yj , 2 ) if j large.
Take finite cover of E by 2−1 balls: −1 ∃ y1 ∈ E : B(y1, 2 ) contains infinitely many elements of {xn}.
−1 −2 Take finite cover of B(y1, 2 ) ∩ E by 2 balls: −1 −2 ∃ y2 ∈ B(y1, 2 ) ∩ E : B(y2, 2 ) contains infinitely many {xn}.
−j −j−1 ∃ yj+1 ∈ B(yj , 2 ): B(yj+1, 2 ) ⊃ infinitely many {xn}. −j The sequence {yj } is Cauchy since ρ(yj+1, yj ) ≤ 2 .
By completeness of E, yj → x for some x ∈ E.
Hart Smith 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: n If not totally bounded, ∃ r > 0 : E 6⊂ ∪j=1B(xj , r) for any {xj }.
Choose x1 ∈ E
Choose x2 ∈ E : x2 ∈/ B(x1, r) ··· n Choose xn ∈ E : xn+1 ∈/ ∪j=1B(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 ⊂ ∪αOα
then there exists r > 0 : ∀x ∈ E , B(x, r) ⊂ Oα for some α.
−n Suppose not: take xn ∈ E such that B(xn, 2 ) 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 Take n such that 2−n < , and ρ(x , x) < . 2 n 2 −n Then B(xn, 2 ) ⊂ B(x, r) ⊂ Oα , ⇒⇐
Hart Smith 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:
π π 1-1, onto Consider the map: tan x :(− 2 , 2 ): ←→ R open intervals ↔ open intervals ⇒ open sets ↔ open sets.
π π (− 2 , 2 ) 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| π π on (− 2 , 2 ) are not equivalent, but give the same collection of open sets.
Hart Smith Math 524