Metric Spaces Flashcards
Metric
For a set M, d: MxM → ℝ is a metric if ∀ x,y,z ∈ M
i) Positivity: d(x,y) ≥ 0, and d(x,y) = 0 ⇔ x = y
ii) Symmetry: d(x,y) = d(y,x)
iii) Triangle Inequality: d(x,y) ≤ d(x,z) + d(z,y)
(M,d) is a metric space.
Norm
A function ‖•‖: M → ℝ is a norm if ∀ x,y ∈ M, ∀ λ ∈ ℝ i) ‖x‖ ≥ 0 and ‖x‖ = 0 ⇔ x = 0, ii) ‖λx‖ = |λ|⋅‖x‖, iii) ‖x + y‖ ≤ ‖x‖ + ‖y‖. (M,‖•‖) is a normed vector space.
Discrete metric
For M set, d: M×M → ℝ is a discrete metric if
d(x,y) = {0 if x = y,
1 if x ≠ y
French railway metric
For a normed vector space (M, ‖•‖), d: M×M → ℝ is a French railway metric if
d(x,y) = {‖x - y‖ if x,y,0 collinear, ‖x‖ + ‖y‖ otherwise
Open ball centre a, radius r
Given a metric space (M,d), a ∈ M, r > 0,
B(a,r) = {x ∈ M: d(x,a) < r} is the open ball with centre a, radius r.
Open set (MS)
For a metric space (M,d), U ⊂ M is open if ∀ x ∈ U, ∃ ε > 0 s.t. B(x, ε) ⊂ U.
Closed set (MS)
For a metric space (M, d), F ⊂ M is closed if M\F is open
DeMorgan’s laws
X\∪(Aᵢ) = ∩(X\Aᵢ)
X\ ∩(Aᵢ) = ∪(X\Aᵢ)
Convergence of a sequence xₙ in metric space M
Let (M, d) be a metric space, (xₙ) ⊂ M a sequence.
xₙ is convergent in M if ∃ L ∈ M s.t. ∀ε > 0, ∃ N ≥ 1 s.t. if n ≥ N, d(xₙ, L) < ε.
Call L the limit, write xₙ → L in M as n → ∞
f continuous at a (MS)
Let (M₁, d₁), (M₂, d₂) be metric spaces, f: M₁ → M₂, a ∈ M₁.
f is cts at a if ∀ ε > 0, ∃ δ > 0 s.t. d₁(x, a) < δ ⇒ d₂(f(x), f(a)) < ε.
f continuous (MS)
f: M₁ → M₂ is cts if f is cts at a, ∀ a ∈ M₁.
f Lipschitz
A function f: M₁ → M₂ is Lipschitz if ∃ L > 0 s.t. d₂(f(x), f(y)) ≤ L⋅d₁(x, y) ∀ x,y ∈ M₁.
Inverse image of U under f
Let (M₁, d₁) and (M₂, d₂) be metric spaces, U ⊂ M₂, f: M₁ → M₂. The inverse image of U under f is f⁻¹(U) = {x ∈ M₁ : f(x) ∈ U}.
Homeomorphism (MS)
Let M₁, M₂ be metric spaces, f: M₁ → M₂. f is a homeomorphism if i) f is a bijection, ii) f is cts from M₁ to M₂, iii) f⁻¹ is cts from M₂ to M₁.
Homeomorphic (MS)
if ∃ f: M₁ → M₂ and f is a homeomorphism, then M₁ and M₂ are homeomorphic.
Topological property
P is a topological property of metric spaces if it is preserved by homeomorphism - if M₁ is homeomorphic to M₂ then P holds for M₁ ⇔ P holds for M₂.
Bounded (MS)
For a metric space (M, d), U ⊂ M is bounded if ∃ a ∈ M, r > 0 s.t. M ⊂ B(a, r).
Topology
Given a set X, X ≠ ∅, a topology τ on X is a collection of subsets of X (τ ⊂ 𝒫(X)), s.t. i) X, ∅ ∈ τ ii) U₁, ..., Uₖ ∈ τ ⇒ U₁ ∩ ... ∩ Uₖ ∈ τ iii) Uᵢ ∈ τ, i ∈ I ⇒ ∪_(i ∈ I) Uᵢ ∈ τ. (X, τ) is a topological space.
Open set (TS)
Let (X, τ) be a topological space.
U ⊂ X is open if U ∈ τ.
Closed set (TS)
Let (X, τ) be a topological space, F ⊂ X.
F is closed if X\F is open.
Neighbourhood (TS)
Let (X, τ) be a topological space, x ∈ X, N ⊂ X.
N is a neighbourhood of x if x ∈ N and N is open.
Interior of A ⊂ X in (X, τ)
Let (X, τ) be a top space, A ⊂ X.
The interior of A, denoted Å, is Å = {x ∈ A | ∃ nhd N of x, N ⊂ A}
Closure of A ⊂ X in (X, τ)
Let (X, τ) be a top space, A ⊂ X.
The closure of A, denoted à (A overbar) is à = {x ∈ X | every nhd of x intersects A}
Finite complement topology
Given a set X, we say (X, τ) is a finite complement topology if τ is such that
τ = {A ⊂ X | A = ∅ or X\A is finite}
Boundary of A ⊂ X in (X, τ)
Let (X, τ) be a top space, A ⊂ X.
The boundary of A is
∂A = {x ∈ X : every nhd of X intersects A and X\A}
Facts about Closure, Interior and Boundary of A ⊂ X
A open ⇔ A = Å
A closed ⇔ A = Ã
∂A = Ã \ Å ∂A = ∂(X\A) Ã = A ∪ ∂A
Clos/u/re, I/n/terior
(A ∪ B)~ = Ã ∪ B~
(A ∩ B)° = A° ∩ B°
Continuity and Preimages
f: (M _1, d_1) → (M₂, d₂), is continuous iff f ⁻¹(U) is open for any U ⊂ M₂ open
Hausdorff space (X, τ)
∀ x,y ∈ X, we can find nhds of U and V of x and y respectively such that U ∩ V = ∅
Convergence in a metric space and open sets
Let (M,d) be a metric space, x_n ∈ M, x ∈ M.
x_n → x ⇔ ∀ U ⊂ M with x ∈ X, ∃ N ≥ 1 s.t. if n ≥ N, x_n ∈ U.
Convergence of a sequence xₙ in a top. space (X, τ)
xₙ converges to L (xₙ → L as n → ∞) if ∀ nhd U of L, ∃ N ≥ 1 s.t. n ≥ N ⇒ xₙ ∈ U
Cover 𝒪 of X
𝒪 ⊂ 𝒫(X) is a cover of X if ∀ x ∈ X, ∃ A ∈ 𝒪 s.t. x ∈ A
Basis ℬ of X
ℬ ⊂ 𝒫(X) is a basis of a topology in X if:
i) ℬ is a cover of X
ii) If B₁, B₂ ∈ ℬ and x ∈ B₁ ∩ B₂, then ∃ B₃ ∈ ℬ s.t. x ∈ B₃ ⊂ B₁ ∩ B₂
Topology τ generated by ℬ
Given a basis ℬ for a topology on X,
τ = {U ⊂ X : ∀ x ∈ U, ∃ B ∈ ℬ s.t. x ∈ B ⊂ U} is the topology generated by ℬ.
f continuous (TS)
f: X₁ → X₂ is cts if f⁻¹(U) is open in X₁, ∀ U open in X₂.
OR f⁻¹(F) closed in X₁ ∀ F closed in X₂
OR f⁻¹(B) open in X₁ ∀ basis elt B ⊂ X₂.
Finer/Coarser topology
If the identity map is cts from (X, τ₁) to (X, τ₂), then τ₂ ⊂ τ₁, so τ₁ is finer than τ₂, or τ₂ is coarser than τ₁.
Homeomorphism (TS)
f: X → Y is a homeomorphism if
i) f is a bijection
ii) f is cts
iii) f⁻¹ is cts
OR f is a bijection and f(U) open in Y ⇔ U open in X
OR f is a bijection and f(F) closed in Y ⇔ F closed in X
Box topology
Given X₁, X₂, … top. spaces, X = X₁×X₂×…, the box topology is τ generated by
ℬ = {∏ᵢ₌₁ ^∞ Uᵢ, Uᵢ open in Xᵢ}.
Product topology
Given X₁, X₂, … top. spaces, X = X₁×X₂×…, the product topology is τ generated by
ℬ = {∏ᵢ₌₁ ^∞ Uᵢ, Uᵢ ⊂ Xᵢ open, ∀ i ≥ k Uᵢ = Xi for some k}
Finite subcover of 𝒪
A cover 𝒪 has finite subcover if ∃ k ≥ 1, A₁, …, Aₖ ∈ 𝒪 s.t. {A₁,…,Aₖ} is still a cover (i.e. X = A₁ ∪ … ∪ Aₖ)
Open cover of (X, τ)
𝒪 is an open cover of X if 𝒪 is a cover of X and 𝒪 ⊂ τ
Compact
A set X is compact if every open cover has a finite subcover
Cover of subspace Y ⊂ X
𝒪 ⊂ 𝒫(X) covers Y ⊂ X if Y ⊂ ∪_{A ∈ 𝒪} A
Uniform continuity of f (MS)
f: (X₁, d₁) → (X₂, d₂) is uniformly cts if ∀ ε > 0 ∃ δ > 0 s.t. ∀ x, y ∈ X₁,
d₁(x, y) < δ ⇒ d₂(f(x), f(y)) < ε.
Sequentially compact (TS)
(X, τ) is sequentially compact if ∀ xₙ ∈ X, ∃ nₖ ≥ 1, nₖ₊₁ > nₖ and y ∈ X s.t. x_{nₖ} → y as k → ∞
Separation A,B of X
A,B ⊂X are a separation of X if
i) A,B ≠ ∅,
ii) A ∩ B = ∅, A ∪ B = X,
iii) A,B open.
X connected
X is connected if there is no possible separation of X.
