Definitions

Question 1

Metric space, metric, distance

Answer 1

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.

Question 2

Limsup of a sequence of real numbers

Answer 2

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

Question 3

Liminf of a sequence of real numbers

Answer 3

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

Question 4

Limsup of a sequence of sets

Answer 4

Let {Eₖ}ₖ₌₁᪲ be a sequence of sets. The limit superior is the following:
∞ ∞
lim sup Eₖ = ⋃ ⋂ Eₙ
k→∞ k=1 n=k

Question 5

Liminf of a sequence of sets

Answer 5

Let {Eₖ}ₖ₌₁᪲ be a sequence of sets. The limit inferior is the following:
∞ ∞
lim inf Eₖ = ⋂ ⋃ Eₙ
k→∞ k=1 n=k

Question 6

Ceiling function

Answer 6

⌈.⌉ : R → Z, x → the least integer ≥ x

Question 7

Floor function

Answer 7

⌊.⌋ : R → Z, x → the greatest integer ≤ x

Question 8

Countable set

Answer 8

A set X is countable if it is either

1) finite, or
2) Can be put into a bijection with N

Question 9

Uncountable set

Answer 9

A set X is uncountable if it is not countable.

Question 10

Neighbourhood, radius

Answer 10

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.

Question 11

Limit point

Answer 11

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.

Question 12

Isolated point

Answer 12

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.

Question 13

Closed set

Answer 13

A subset E of a metric space is closed if every limit point of E is a point of E.

Question 14

Interior point

Answer 14

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.

Question 15

Open set

Answer 15

A set E in a metric space is open if every point of E is an interior point in E.

Question 16

Perfect set

Answer 16

A set E in a metric space is perfect if E is closed and every point of E is a limit point of E.

Question 17

Bounded set

Answer 17

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.

Question 18

Dense set

Answer 18

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).

Question 19

Closure of a set

Answer 19

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’

Question 20

Open cover

Answer 20

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

Question 21

Compact set

Answer 21

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

Question 22

Lipschitz (continuous), Lipschitz constant

Answer 22

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.

Question 23

Bilipschitz (continuous)

Answer 23

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.

Question 24

Hausdorff metric, 𝛅-parallel body

Answer 24

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}

Question 25

𝜎 - algebra

Answer 25

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.

Question 26

Borel 𝜎 - algebra

Answer 26

Let X be a metric space. The Borel 𝜎 - algebra of X is the smallest 𝜎 - algebra containing all open subsets of X.

Question 27

Measure

Answer 27

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

Question 28

Measure space

Answer 28

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.

Question 29

Null set

Answer 29

Let (X, ℳ, 𝜇) be a measure space. A set E ϵ ℳ is called a null set if 𝜇(E) = 0.

Question 30

Almost everywhere

Answer 30

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.

Question 31

Complete measure

Answer 31

Let (X, ℳ, 𝜇) be a measure space. A measure whose domain includes all the subsets of null sets is called complete.

Question 32

Support of a measure

Answer 32

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.

Question 33

Outer measure

Answer 33

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

Question 34

𝜇*-measurable

Answer 34

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.

Question 35

Metric outer measure

Answer 35

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.

Question 36

Cantor set

Answer 36

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.

Question 37

N_𝛿(F)

Answer 37

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.

Question 38

Box dimension

Answer 38

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

Question 39

𝛿-cover of F

Answer 39

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.

Question 40

Hausdorff measure

Answer 40

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

Question 41

s-dimensional Hausdorff measure

Answer 41

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

Question 42

Hausdorff dimension

Answer 42

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.

Question 43

Full Hausdorff dimension

Answer 43

Let F ⊂ Rⁿ. F has full Hausdorff dimension in Rⁿ if dim_H (F) = n.

Question 44

Thick subset

Answer 44

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.

Question 45

Contraction, similarity

Answer 45

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.

Question 46

Iterated Function System

Answer 46

A family of contractions {S₁, …, Sₘ} is called an iterated function system (IFS).

Question 47

IFS fractal

Answer 47

An invariant set for an IFS is called an IFS fractal.

Question 48

Continuous functions on metric spaces

Answer 48

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.

Question 49

Self-similar set

Answer 49

The invariant set of an IFS on Rⁿ consisting only of similarities is called a self-similar set.

Question 50

Open set condition

Answer 50

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

Question 51

(𝛼, 𝛽, S)-game

Answer 51

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.

Question 52

Schmidt games

Answer 52

The collection of (𝛼, 𝛽, S)-games are referred to as Schmidt games.

Question 53

Player A wins

Answer 53

Let S be a subset of Rⁿ. The second player, Player A, wins if the intersection of these balls lies in S.

Question 54

(𝛼, 𝛽)-winning sets

Answer 54

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.

Question 55

𝛼-winning sets

Answer 55

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.

Question 56

Winning set

Answer 56

Let S be a subset of Rⁿ. S is called winning if it is 𝛼-winning for some 0 < 𝛼 < 1.

Question 57

(𝛼, 𝛽, S)-winning strategy

Answer 57

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.

Question 58

Well-approximable numbers, WA

Answer 58

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.

Question 59

Badly-approximable numbers, BA

Answer 59

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.

Question 60

Lowest terms

Answer 60

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.