Definitions

Question 1

A metric space (X, d)

Answer 1

Consists of a non-empty set X and a non-negative real valued metric d: X × X → R>

Satisifes the axioms:

1. d(x, y) = 0 ⇐⇒ x = y for all x, y ∈ X
2. d(x, y) = d(y, x) for all x, y ∈ X
3. d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z ∈ X (triangle inequality)

Known informally as a distance function .

Question 2

Subspace

Answer 2

Given any subset W ⊆ X, the restriction of d to W determines the subspace (W, d := d|W ) of (X, d)

Question 3

Open and closed balls

Answer 3

For any metric space (X, d), the open ball and closed ball of radius r > 0 around a point x ∈ X are the subspaces:

Br(x) := {y : d(y, x) < r} and

_
Br(x) := {y : d(y, x) ≤ r} ⊆ X

respectively.

Question 4

Euclidean n-space (R^n, d2)

Answer 4

Euclidean n-space (R^n, d2) consists of all real n-dimensional vectors:

x = (x1, . . . , xn)

Question 5

Isometry

Answer 5

For any two metric spaces (X, dX) and (Y, dY ), a Bijection
f : X → Y is an isometry whenever:

dX(x, y) = dY (f(x), f(y)) for all x, y ∈ X

Question 6

A graph Γ := (V, E)

Answer 6

A graph Γ := (V, E) consists of a set V of vertices , and a set E of edges.

Each edge vw ∈ E may be interpreted as joining two vertices v, w ∈ V

Question 7

A path in Γ

The length of a path

Answer 7

A path in Γ from u to w is a finite sequence of edges:

π(u, w) = (uv_1, v_1v_2, . . . , v_(n−2)v_(n−1), v_(n−1)w)

Such a path has length `(π(u, w)) = n

Question 8

Path connected

Answer 8

A graph is path connected whenever there is a path joining any pair of vertices

Question 9

Edge metric e

Answer 9

The edge metric e on the vertex set V of a path connected graph is defined by:

e(u, w) = min_(π(u,w)) L(π(u, w))

Question 10

Alphabet

Answer 10

An alphabet is a finite set A of letters

Question 11

Word

Answer 11

A finite sequence of letters is a word in A; if A := {a, b, c, o, t}, then words include act, boat, and obbbt

Question 12

Word graph

Answer 12

The vertex set W of the associated word graph Γ(A)
consists of all possible words in A.

Words w1 and w2 are joined by an edge iff they differ by one of:

1. inserting or deleting a letter
2. swapping two adjacent letters
3. replacing one letter with another

Question 13

Word Metric

Answer 13

The word metric dw on W is the edge metric on the associated word graph.

ex:
dw(act,boat) = 3
act, bact, baot, boat

Question 14

Binary sequences

Answer 14

X = {0, 1}^∞ is the set of all infinite binary sequences x = x_0x_1 . . . ,

x_n = 0 or 1 for all n ≥ 0

x = 1100010 . . . is a typical element

Question 15

Bounded

Answer 15

A real-valued function f on a closed interval [a, b] ⊂ R is bounded whenever
there exists a constant K such that |f(x)| ≤ K for every x ∈ [a, b]

Question 16

Real polynomial of degree n

Answer 16

a(x) = a_nx^n + · · · + a_1x + a_0

Question 17

Cartesian product of two metric spaces (X, d) and (X’,d’)

Answer 17

The cartesian product of two metric spaces (X, d) and (X’,d’) is the set X × X’
equipped with one of the metrics:

1. d_a((x, x’),(y, y’)) = d(x, y) + d’(x’, y’)
2. d_b((x, x’),(y, y’)) = [ d(x, y)^2 + d’(x’, y’)^2 ]^1/2
3. d_c((x, x’),(y, y’)) = max{d(x, y), d’(x’, y’)}

for any x, y ∈ X and x’, y’ ∈ X’
.

Question 18

Lipschitz equivalent

Answer 18

Two metrics d and e on a given set X are Lipschitz equivalent whenever there exist positive constants h, k ∈ R such that:

he(x, y) ≤ d(x, y) ≤ ke(x, y)

for every x, y in X

Question 19

An interior point U ⊆ X

Answer 19

An interior point u ∈ U is one for which there exists e > 0 such that B_e(u) ⊆ U

Question 20

The interior of U

Answer 20

The interior of U is the subset U◦ ⊆ U of all interior points

Question 21

U is open in X

Answer 21

U◦ = U

Question 22

A closure point of U ⊆ X

Answer 22

A point x ∈ X is a closure point of U ⊆ X if B_e(x) ∩ U is

non-empty for every u > 0; If U = U, then U is closed in X

Question 23

The closure of U

Answer 23

The closure of U is the superset:
_
U ⊇ U of all closure points

Question 24

U is closed in X

Answer 24

_

U = U

Question 25

A partially open ball in X

Answer 25

A set Pr(x):= Br(x) ∪ P, where P is a proper subset of {p : d(x, p) = r}

Question 26

A sequence (x_n) converges

Answer 26

A sequence (x_n) converges to the point x ∈ X whenever
∀e > 0, ∃N ∈ N such that n ≥ N ⇒ d(x, x_n) < e;
in this situation, x is known as the limit of (x_n)

∀e > 0, ∃N ∈ N such that n ≥ N ⇒ x_n ∈ B_e(x).

Question 27

Cauchy sequence

Answer 27

In any metric space (X, d), a Cauchy sequence (x_n) satisfies ∀e > 0, ∃N ∈ N such that m, n ≥ N ⇒ d(x_m, x_n) < e

Question 28

Dense

Answer 28

A subset Y is dense in (X, d) whenever Y = X

Question 29

Bounded

Answer 29

A subset A of a metric space (X, d) is bounded whenever there exists x_0 ∈ X and M ∈ R such that d(x, x_0) ≤ M for every x ∈ A
A function f : S → X is bounded whenever its image f(S) ⊂ X is a bounded, for any set S

Question 30

Diameter

Answer 30

The diameter diam(A) of a bounded non-empty subset A ⊆ X is the real number sup{d(x, y) : x, y ∈ A}

Question 31

Boundary point

Answer 31

A boundary point x ∈ X of A is one for which every open ball B_e(x) meets both A and X \ A; the boundary ∂A of A is the set of all such boundary points

Question 32

The boundary

Answer 32

The boundary ∂A of A is the set of all such boundary points