Definitions Flashcards
Metric space, metric
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.
Taxi cab metric
Define the taxi cab metric d’ on Rⁿ to be
n
d’(x, y) = ∑ |xⱼ - yⱼ| for x, y ∈ Rⁿ
j=1
Discrete metric
For any set X, define the discrete metric as
d(x, y) = 1 if x ≄ y
0 if x = y
Subspace of a metric space
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.
Converge, limit in a metric space
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.
Continuous at a, continuous function in a metric space
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.
Open ball
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.
Open, closed subset of a metric space
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.
Topologically equivalent
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.
Lipschitz equivalent
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.
Topological space, topology, open sets
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.
Usual topology
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.
Indiscrete topology
Let X be any set. Then T = {∅, X} is the indiscrete topology on X.
Discrete topology
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.
Cofinite topology
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.
Basis of a topological space
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.
Coarser, finer
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.
Closed in a topological space
A subset A of a topological space X is closed if its complement X\A is open.
Convergence in a topological space
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.
Continuous function between topological spaces
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.
Homeomorphism
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.
Interior
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
Closure
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
Hausdorff space
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 = ∅.
Subspace topology
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.
Product topology
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}.
Open cover
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
Compact
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.
Connected, disconnected space
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.
Connected subset
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.
Partition
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 = ∅.
Connected component
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.
Path, path connected
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.
Cauchy sequence
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.
Complete metric space
A metric space (X, d) is complete if every Cauchy sequence in X converges to an element of X.
Fixed point
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.
Contraction
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
Null set
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.
Outer measure
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
(Lebesgue) measurable
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.
Lebesgue measure
For E ∈ ℳ we define m(E) = m*(E). Then m is the Lebesgue measure on R.
σ-algebra
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.
σ-algebra generated by 𝜀
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).
Borel algebra
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.
Measure, measure space
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.
Finite measure
A measure µ on a set X is a finite measure if µ(X) < ∞.
Probability measure
A measure µ on a set X is a probability measure if µ(X) = 1.
