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}
