Definitions Flashcards
Metric space, metric, distance
A set X is called a metric space if there exists a function
d: X x X → [0, ∞)
such that
(a) d(x, y) > 0 if x ≄ y; d(x, x) = 0;
(b) d(x, y) = d(y, x)
(c) d(x, y) ≤ d(x, z) + d(z, y) for any z ∈ X (triangle inequality)
The function d is the metric or distance function, and the number d(x, y) is the distance between x and y.
Limsup of a sequence of real numbers
Let {xₖ}ₖ₌₁᪲, be a sequence of real numbers. The limit superior of the sequence is
lim sup xₖ = inf ( sup xₙ )
k→∞ k≥1 n≥k
Liminf of a sequence of real numbers
Let {xₖ}ₖ₌₁᪲ be a sequence of real numbers. The limit inferior of the sequence is
lim inf xₖ = sup ( inf xₙ )
k→∞ k≥1 n≥k
Limsup of a sequence of sets
Let {Eₖ}ₖ₌₁᪲ be a sequence of sets. The limit superior is the following:
∞ ∞
lim sup Eₖ = ⋃ ⋂ Eₙ
k→∞ k=1 n=k
Liminf of a sequence of sets
Let {Eₖ}ₖ₌₁᪲ be a sequence of sets. The limit inferior is the following:
∞ ∞
lim inf Eₖ = ⋂ ⋃ Eₙ
k→∞ k=1 n=k
Ceiling function
⌈.⌉ : R → Z, x → the least integer ≥ x
Floor function
⌊.⌋ : R → Z, x → the greatest integer ≤ x
Countable set
A set X is countable if it is either
1) finite, or
2) Can be put into a bijection with N
Uncountable set
A set X is uncountable if it is not countable.
Neighbourhood, radius
Let X be a metric space. A neighbourhood of a point p in X is a set
B(p, r) := {q ϵ X : d(p, q) < r}
The number r > 0 is the radius. This is also the definition of an open ball around the point p of radius r.
Limit point
A point p in a metric space X is a limit point of a subset E of X if every neighbourhood of p contains a point q ≠ p with q in E.
Isolated point
If p ϵ E, a subset of a metric space X, and p is not a limit point of E, then p is an isolated point of E.
Closed set
A subset E of a metric space is closed if every limit point of E is a point of E.
Interior point
A point p is an interior point of a subset E of a metric space if there is a neighbourhood N of p such that N is a subset of E.
Open set
A set E in a metric space is open if every point of E is an interior point in E.
Perfect set
A set E in a metric space is perfect if E is closed and every point of E is a limit point of E.
Bounded set
A set E in a metric space X is bounded if there exists a real number M > 0 and a point q ϵ X such that d(q, p) < M for all p ϵ E.
Dense set
A set E in a metric space X is dense in X if every point of X is a limit point of E or a point of E (or both).
Closure of a set
Let X be a metric space and E a subset of X. If E’ denotes the set of all limit points of E in X, then the closure of E is the set
Ē = E ⋃ E’
Open cover
An open cover of a set E in a metric space X is a collection {U𝛼} of open subsets of X such that
E ⊂ ⋃ U𝛼
𝛂
Compact set
A subset K of a metric space X is compact if every open cover of K contains a finite subcover. More explicitly, if {U𝛼} is an open cover of K, then there exists finitely many indices 𝛼₁, …, 𝛼ₙ such that
K ⊂ U𝛼₁ ⋃ … ⋃ U𝛼ₙ
Lipschitz (continuous), Lipschitz constant
Given metric spaces (X, dₓ), (Y, dᵧ), a function f: x → y is called Lipschitz (continuous) if there exists a constant K > 0 such that for all x₁, x₂ ϵ X, we have that
dᵧ (f(x₁,) f(x₂)) ≤ Kdₓ (x₁, x₂)
The constant K is referred to as the Lipschitz constant of f. Note that K suffices for every x₁, x₂ ϵ X.
Bilipschitz (continuous)
Given metric spaces (X, dₓ), (Y, dᵧ), a function f: x → y is called bilipschitz (continuous) if there exists a constant K ≥ 1 such that for all x₁, x₂ ϵ X, we have that
1/K dₓ (x₁, x₂) ≤ dᵧ (f(x₁,) f(x₂)) ≤ Kdₓ (x₁, x₂)
The constant K is referred to as the biipschitz constant of f. Note that K suffices for every x₁, x₂ ϵ X.
Hausdorff metric, 𝛅-parallel body
Let D be a closed subset of Rⁿ and let 𝒮 := 𝒮(D) denote the collection of non-empty compact subsets of D. Let 𝛅 > 0. For any A ϵ 𐊖, define the 𝛅-parallel body to A as the following set:
A_𝛅 = {x ϵ D : |x-a| ≤ 𝛅 for some a ϵ A}
For any A, B ϵ 𝒮, the Hausdorff metric is defined as
d_H (A, B) = inf {𝛅 : A ⊂ B_𝛅 and B ⊂ A_𝛅}
𝜎 - algebra
Let X be a set. A 𝜎 - algebra on X is a non-empty collection A of subsets of X such that
1) (closed under countable unions): If {Eₙ}ₙ₌₁᪲ ⊂ A, then
∞
⋃ Eₙ ϵ A
n=1
and
2) (closed under taking complements): If E ϵ A, then E ͨ ϵ A.
Borel 𝜎 - algebra
Let X be a metric space. The Borel 𝜎 - algebra of X is the smallest 𝜎 - algebra containing all open subsets of X.
Measure
Let X be a set and ℳ be a 𝜎 - algebra on X. A measure on ℳ is a function
𝜇 : ℳ → [0, ∞]
such that
(i) 𝜇(∅) = 0
(ii) (Countable additivity): If {Eₙ}ₙ₌₁᪲ is a sequence of disjoint sets in ℳ, then
∞ ∞
𝜇( ⋃ Eₙ ) = ∑ 𝜇(Eₙ)
n=1 n=1
Measure space
If X is a set, ℳ is a 𝜎 - algebra on X, and 𝜇 is a measure on ℳ, then (X, ℳ, 𝜇) is called a measure space.
Null set
Let (X, ℳ, 𝜇) be a measure space. A set E ϵ ℳ is called a null set if 𝜇(E) = 0.
Almost everywhere
Let (X, ℳ, 𝜇) be a measure space. If a statement about points of X is true except for x in some null set, we say that it is true almost everywhere, or 𝜇-almost everywhere.
Complete measure
Let (X, ℳ, 𝜇) be a measure space. A measure whose domain includes all the subsets of null sets is called complete.
Support of a measure
Let X be a metric space, ℳ a 𝜎 - algebra containing the Borel 𝜎 - algebra, and 𝜇 a measure on ℳ. Let N be the union of all open sets U ⊂ X such that 𝜇(U) = 0. The support of the measure, denoted supp(𝜇), is the complement of N.
Note that supp(𝜇) is a closed subset of X.
Outer measure
Let X be a non-empty set and 𝒫(X) denote the collection of all subsets of X. An outer measure on X is a function
𝜇* : 𝒫(X) → [0, ∞]
that satisfies
(i) 𝜇*(∅) = 0
(ii) 𝜇(A) ≤ 𝜇(B) if A ⊂ B
(iii)
∞ ∞
𝜇( ⋃ Aₙ ) ≤ ∑ 𝜇(Aₙ)
n=1 n=1
𝜇*-measurable
Let X be a non-empty set and 𝜇* an outer measure on X. A set A ⊂ X is 𝜇*-measurable if
𝜇(E) = 𝜇(E ⋂ A) + 𝜇*(E ⋂ Aᶜ)
for all E ⊂ X.
Metric outer measure
Let (X, d) be a metric space. An outer measure 𝜇* on X is called a metric outer measure if
𝜇(A ⋃ B) = 𝜇(A) + 𝜇*(B)
whenever inf{ d(a, b) : a ϵ A and b ϵ B} > 0.
Cantor set
The intersection over all nested sets E₀ ⊃ E₁ ⊃ E₂ ⊃… where E₀ = [0, 1], E₁ = [0, 1/3] U [2/3, 1] etc, so Eₙ is the disjoint union of 2ⁿ closed intervals, each of length 3⁻ⁿ, is called the Cantor set.
N_𝛿(F)
Let F be a bounded, non-empty subset of Rⁿ and let 𝛿 > 0. Define
N_𝛿(F) = smallest number of sets of diameter ≤ 𝛿 that cover F.
Box dimension
The lower and upper box dimensions of a bounded, nonempty subset F of Rⁿ are respectively
lower dimᵦF = lim inf (log N_𝛿(F)) / -log𝛿
𝛿 → 0
upper dimᵦF = lim sup (log N_𝛿(F)) / -log𝛿
𝛿 → 0
If the lower and upper box dimensions agree, then the box dimension of F is
dimᵦF = lim (log N_𝛿(F)) / -log𝛿
𝛿 → 0
𝛿-cover of F
Let F ⊂ Rⁿ and 𝛿 > 0. If {Uⱼ} (j = 1 to ∞) is a countable collection of sets of Rⁿ such that
∞
F ⊂ ⋃ Uⱼ with 0 < |Uⱼ| ≤ 𝛿 for each j
j=1
we say that {Uⱼ} (j = 1 to ∞) is a 𝛿-cover of F.
Hausdorff measure
Let s be a non-negative real number, and F a subset of Rⁿ. We define
∞
ℋˢ𝛿 (F) := inf{∑ |Uⱼ|ˢ : {Uⱼ}(j = 1 to ∞) is a 𝛿-cover of F}
j=1
s-dimensional Hausdorff measure
Let s ≥ 0 and F ⊂ Rⁿ. The s-dimensional Hausdorff measure of F is
ℋ^s (F) := lim ℋ_𝛿 ^s (F)
𝛿 → 0
Note that this is an outer measure
Hausdorff dimension
Let F be a subset of Rⁿ. The Hausdorff dimension of F is
dim_H (F) := inf{s : ℋ^s (F) = 0} = sup{s : ℋ^s (F) = ∞},
except when the latter set is the empty set, in which case we set the supremum to be 0.
Full Hausdorff dimension
Let F ⊂ Rⁿ. F has full Hausdorff dimension in Rⁿ if dim_H (F) = n.
Thick subset
Let F ⊂ Rⁿ. We say F is a thick subset in Rⁿ if dim_H (F ⋂ U) = n for every non-empty open subset of Rⁿ.
Contraction, similarity
A mapping S: D → D is called a contraction if there exists a constant 0 < c < 1 such that
|S(x) - S(y)| ≤ c|x − y|
for all x, y ϵ D. If equality holds for all x, y ϵ D, then the mapping is called a similarity.
Iterated Function System
A family of contractions {S₁, …, Sₘ} is called an iterated function system (IFS).
IFS fractal
An invariant set for an IFS is called an IFS fractal.
Continuous functions on metric spaces
Let (X, dₓ) and (Y, dᵧ) be metric spaces, E ⊂ X, p ϵ E, and f: E → Y a function. Then f is continuous at p if for every 𝜀 > 0 there exists a 𝛿 > 0 such that
dᵧ( f(x), f(p) ) < 𝜀
for all points x ϵ E for which dₓ(x, p) < 𝛿.
If f is continuous at every point of E, then f is said to be continuous on E.
Self-similar set
The invariant set of an IFS on Rⁿ consisting only of similarities is called a self-similar set.
Open set condition
A family {S₁, …, Sₘ} of similarities on Rⁿ satisfies the open set condition if there exists a non-empty, open, bounded set V ⊂ Rⁿ such that
m
⨆ Sⱼ(V) ⊂ V
j = 1
(𝛼, 𝛽, S)-game
Let 0 < 𝛼,𝛽 < 1 be two chosen constants. Let S be a subset of M := Rⁿ and 𝜌(.) denote the radius of a closed ball. The two players, Player A and Player B, alternate choosing nested closed balls
B₁ ⊃ A₁ ⊃ B₂ ⊃ A₂ ⊃ …
on M according to the following rules:
𝜌(Aₙ) = 𝛼𝜌(Bₙ), and 𝜌(Bₙ) = 𝛽𝜌(Aₙ₋₁)
Notice that B₁ is an arbitrary closed ball. This game is called the (𝛼, 𝛽, S)-game.
Schmidt games
The collection of (𝛼, 𝛽, S)-games are referred to as Schmidt games.
Player A wins
Let S be a subset of Rⁿ. The second player, Player A, wins if the intersection of these balls lies in S.
(𝛼, 𝛽)-winning sets
Let S be a subset of Rⁿ. S is called (𝛼, 𝛽)-winning if Player A can always win for the given 𝛼 and 𝛽.
𝛼-winning sets
Let S be a subset of Rⁿ. S is called 𝛼-winning if Player A can always win for the given 𝛼 and every 0 < 𝛽 < 1.
Winning set
Let S be a subset of Rⁿ. S is called winning if it is 𝛼-winning for some 0 < 𝛼 < 1.
(𝛼, 𝛽, S)-winning strategy
Player A has a (𝛼, 𝛽, S)-winning strategy if at each step n of the (𝛼, 𝛽, S)-game, Player A can make a choice of ball Aₙ so that Player A wins. The choice of Aₙ at step n is referred to as Player A’s nth move and the choice of Bₙ is referred to as Player B’s nth move.
Well-approximable numbers, WA
A real number x is well-approximable if for every constant c > 0, there exists a rational number p/q such that
|x - p/q| < c / q^2
Note that the rational number p/q depends on c. Let WA denote the set of well approximable numbers.
Badly-approximable numbers, BA
A real number x is badly-approximable if there exists a constant c > 0 such that for all rational numbers p/q,
|x - p/q| > c / q^2
Note that the constant c depends on x. Let BA denote the set of badly approximable numbers.
Lowest terms
A non-zero rational number p/q is in lowest terms if p and q are relatively prime. We call 0/1 the lowest term form of the number 0 and say that 1 and 0 are relatively prime.
We will use the notation (p, q) = 1 to denote that p and q are relatively prime.
