Definitions

Question 1

Metric space, metric

Answer 1

A metric space (X, d) consists of a non-empty set X and a function d : X x X ⟶ R satisfying the conditions
(M1) d(x, y) ≥ 0 for all x, y ∈ X, and d(x, y) = 0 ⟺ x = y
(M2) Symmetry: d(y, x) = d(x, y) for all x, y ∈ X
(M3) Triangle inequality: d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z ∈ X
The function d is called a metric.

Question 2

Taxi cab metric

Answer 2

Define the taxi cab metric d’ on Rⁿ to be
n
d’(x, y) = ∑ |xⱼ - yⱼ| for x, y ∈ Rⁿ
j=1

Question 3

Discrete metric

Answer 3

For any set X, define the discrete metric as

d(x, y) = 1 if x ≄ y
0 if x = y

Question 4

Subspace of a metric space

Answer 4

A subspace of a metric space (X, dx) is a non-empty subset Y together with the metric dy given by restricting dx to Y,

dy(y, y’) = dx(y, y’) for all y, y’ ∈ Y.

(Y, dy) is a metric space.

Question 5

Converge, limit in a metric space

Answer 5

Let (X, d) be a metric space, and let (aₙ) be a sequence of points in X. For a point a ∈ X, we say aₙ converges to a, written aₙ ⟶ a as n ⟶ ∞ if, for any real number 𝜀 > 0, there exists some N ∈ N with d(aₙ, a) < 𝜀 for all n > N. We also say that a is the limit of the sequence.

Question 6

Continuous at a, continuous function in a metric space

Answer 6

Let (X, dx) and (Y, dy) be metric spaces, and let f : X ⟶ Y. For a ∈ X, we say f is continuous at a if, given 𝜀 > 0, there exists 𝛿 > 0 so that dy(f(x), f(a)) < 𝜀 for all x ∈ X with dx(x, a) < 𝛿. We say f is continuous if it is continuous at all a ∈ X.

Question 7

Open ball

Answer 7

Let (X, d) be a metric space. For any real number 𝜀 > 0 and any a ∈ X, the set

B𝜀(a) = {x ∈ X : d(x, a) < 𝜀}

is called the open ball in X of radius 𝜀, centred at a.

Question 8

Open, closed subset of a metric space

Answer 8

A subset U of a metric space is open if, for every x ∈ U, there is some 𝜀 > 0 such that B𝜀(x) ⊆ U. (Note that 𝜀 can depend on x.)
A subset V of X is closed if its complement X\V is open.

Question 9

Topologically equivalent

Answer 9

Let d1 and d2 be two metrics on the same set X. We say that d1 and d2 are topologically equivalent if the open sets with respect to d1 are the same as the open sets with respect to d2.

Question 9

Lipschitz equivalent

Answer 9

Let d1 and d2 be two metrics on the set X. We say that d1 and d2 are Lipschitz equivalent if there are constants A ≥ B > 0 such that
Bd1(x, y) ≤ d2(x, y) ≤ Ad1(x, y)
for all x, y ∈ X.

Question 10

Topological space, topology, open sets

Answer 10

A topological space (X, T) is a non-empty set X together with a family T of subsets X satisfying the following conditions:
(T1) X ∈ T and ∅ ∈ T
(T2) If U1, U2 ∈ T then U1 ⋂ U2 ∈ T
(T3) If Ui ∈ T are any collection of sets in T, indexed by i ∈ I for some set I, then
⋃ Ui ∈ T
iϵI
We call a collection T of subsets satisfying (T1) - (T3) a topology on X, and we call the elements of T to open sets of X in the topology T.

Question 11

Usual topology

Answer 11

Let (X, d) be any metric space and let T be the collection of open sets defined with respect to the Euclidean metric on R, Rⁿ, C, etc. This is the usual topology on these spaces.

Question 12

Indiscrete topology

Answer 12

Let X be any set. Then T = {∅, X} is the indiscrete topology on X.

Question 13

Discrete topology

Answer 13

Let X be a non-empty set, and let T be the collection of all subsets of X. This is the discrete topology on X.

Question 14

Cofinite topology

Answer 14

Let X be a non-empty set and let T consist of all subsets U ⊆ X whose complement X\U is finite, together with the empty set. Then T is the cofinite topology on X.

Question 15

Basis of a topological space

Answer 15

Given a topological space (X, T), a basis of T is a subset B of T such that every open set is a union of sets from B.

Question 16

Coarser, finer

Answer 16

Let T1, T2 be two topologies on X. We say T1 is coarser than T2 (or weaker than T2) if every open set of T1 is also an open set in T2 (i.e. T1 ⊆ T2). We also say that T2 is finer than T1.

Question 17

Closed in a topological space

Answer 17

A subset A of a topological space X is closed if its complement X\A is open.

Question 18

Convergence in a topological space

Answer 18

Let aₙ, n ≥ 1, be a sequence of points in a topological space X. We say that aₙ converges to a point a ∈ X, written aₙ ⟶ a as n ⟶ ∞, if, for every open set U of X with a ∈ U, there is some N ∈ N such that aₙ ∈ U for all n > N.

Question 19

Continuous function between topological spaces

Answer 19

A function f : X ⟶ Y between topological spaces is continuous if, for every open set U of Y, the subset f⁻¹(U) is an open subset of X.

Question 20

Homeomorphism

Answer 20

A homeomorphism between topological spaces X and Y is a continuous function f : X ⟶ Y which is bijective and whose inverse function f⁻¹ : Y ⟶ X is also continuous. We say X and Y are homeomorphic if there is a homeomorphism between them.

Question 21

Interior

Answer 21

Let X be a topological space. For any A ⊆ X, the interior of A, written A°, is the union of all open subsets of X contained in A:

A° = ⋃ U
U open:
U ⊆ A

Question 22

Closure

Answer 22

Let X be a topological space. For any A ⊆ X, the closure of A, written Aࠡ, is the intersection of all closed subsets of X which contain A:

Aࠡ = ⋂ C
C closed;
A ⊆ C

Question 23

Hausdorff space

Answer 23

A topological space is Hausdorff if, given any points x, y ∈ X with x ≄ y, there exist open sets U, V in X with x ∈ U, y ∈ V and U ⋂ V = ∅.

Question 24

Subspace topology

Answer 24

If (X, T) is a topological space and A is any non-empty subset of X, then we define T_A = {U ⋂ A : U ∈ T}. Thus a subset V of A is open in A iff there exists an open set U with V = U ⋂ A. We call T_A the subspace topology on A induced by X.

Question 25

Product topology

Answer 25

Let X, Y be topological spaces. The product topology on X x Y is the topology with basis B = {U x V : U open in X, V open in Y}.

Question 26

Open cover

Answer 26

Let X be a topological space and let A be any subset of X. An open cover of A in X is a family of open sets Ui, i ∈ I, such that
A ⊆ ⋃ Ui
i ϵ I

Question 27

Compact

Answer 27

Let X be a topological space and let A be any subset of X. A is compact if every open cover Ui, i ∈ I has a finite subcover, i.e. there are i1, …, in ∈ I for some n ∈ N with A ⊆ U_i1 ⋃ … ⋃ U_in.

Question 28

Connected, disconnected space

Answer 28

A topological space X is connected if there is no surjective continuous function f : X ⟶ {0, 1} (where we give the two-point set {0,1} the discrete topology). Otherwise, we say X is disconnected.

Question 29

Connected subset

Answer 29

We say a non-empty subset Y of a topological space X is connected if it is connected as a topological space with its subspace topology induced from X.

Question 30

Partition

Answer 30

A partition of a topological space X is a pair of non-empty open subsets A, B such that A ⋃ B = X and A ⋂ B = ∅.

Question 31

Connected component

Answer 31

Let X be a topological space and let x ∈ X. Then the connected component Cx of x in X is the union of all connected subsets of X containing x:

Cx = ⋃ V
xϵV⊆X
V connected

Then V is connected, so Cx is the unique largest connected subset of X containing x.

Question 32

Path, path connected

Answer 32

A topological space X is path connected if, for any x, y ∈ X, there is a continuous function p : [0,1] ⟶ X with p(0) = x and p(1) = y. We call p a path from x to y.

Question 33

Cauchy sequence

Answer 33

A sequence (aₙ) n≥1 in a metric space (X, d) is a Cauchy sequence if, for every 𝜀 > 0, there is an N ∈ N with d(aₘ, aₙ) < 𝜀 for all m, n > N.

Question 34

Complete metric space

Answer 34

A metric space (X, d) is complete if every Cauchy sequence in X converges to an element of X.

Question 35

Fixed point

Answer 35

Let S be any set and f : S ⟶ S a function. A fixed point of f is a point s ∈ S with f(s) = s.

Question 36

Contraction

Answer 36

Let (X, d) be a metric space. A function f : X ⟶ X is a contraction if there is some K < 1 such that

d( f(x), f(y) ) ≤ Kd(x, y) for all x, y ∈ X

Question 37

Null set

Answer 37

A subset A of R is a null set if, given 𝜀 > 0, there is a countable family of open intervals In, n ≥ 1 such that
∞ ∞
A ⊆ ⋃ In and ∑ m(In) < 𝜀
n=1 n=1
where, for an open interval I = (u, v) with u ≤ v, we define m(I) = v - u.

Question 38

Outer measure

Answer 38

The outer measure of A is m*(A) = inf Z(A), where
∞ ∞
Z(A) := {∑ m(Iₙ) : I₁, I₂,… are open intervals with A ⊆ ⋃ Iₙ}
n=1 n=1

Question 39

(Lebesgue) measurable

Answer 39

We say E ⊆ R is (Lebesgue) measurable if, for all A ⊆ R,
m(A) = m(A ⋂ E) + m*(A ⋂ E ͨ ).
We write ℳ for the collection of measurable subsets of R.

Question 40

Lebesgue measure

Answer 40

For E ∈ ℳ we define m(E) = m*(E). Then m is the Lebesgue measure on R.

Question 41

σ-algebra

Answer 41

A σ-algebra on a set X is a family of subsets of X such that
(i) ∅ ∈ B
(ii) if A ∈ B then X\A ∈ B
(iii) if A1, A2, … is a countable sequence of sets in B, then
∞
⋃ An ∈ B
n=1

Thus a family of subsets of X is a σ-algebra if it contains ∅ and is closed under taking complements and countable unions.

Question 42

σ-algebra generated by 𝜀

Answer 42

Let 𝜀 be any family of subsets of a set X. Then the σ-algebra generated by 𝜀 is the smallest σ-algebra containing 𝜀 (i.e. the intersection of all σ-algebras containing 𝜀).

Question 43

Borel algebra

Answer 43

Let (X, T) be a topological space. The Borel algebra on X is the σ-algebra generated by T. Its elements are called Borel subsets of X.

Question 44

Measure, measure space

Answer 44

Let X be a set and B a σ-algebra on X. A measure is a function µ : B ⟶ [0, ∞] such that
(i) µ(∅) = 0
(ii) for any countable infinite family of sets A1, A2, … ∈ B with Ai ⋂ Aj = ∅ for i ≄ j, we have
∞ ∞
µ ( ⋃ An ) = ∑ µ(An)
n=1 n=1

We call the triple (X, B, µ) a measure space.

Question 45

Finite measure

Answer 45

A measure µ on a set X is a finite measure if µ(X) < ∞.

Question 46

Probability measure

Answer 46

A measure µ on a set X is a probability measure if µ(X) = 1.