Theorems Flashcards
Intermediate Value Theorem
Let f : [a, b] ⟶ R be a continuous function. For any v between f(a) and f(b), there is at least once x ∈ [a, b] with f(x) = v.
Bounds of a continuous function [a, b] ⟶ R.
Let f : [a, b] ⟶ R be a continuous function. Then f(x) is bounded and attains its bounds, i.e. f has a finite maximum M and minimum m in [a, b]. More precisely, there are points x_max, x_min ∈ [a, b] so that f(x) ≤ f(x_max) = M and f(x) ≥ f(x_min) = m for all x ∈ [a, b].
When is B_𝜀(a) open?
For any a ∈ X and any 𝜀 > 0, the set B_𝜀(a) is an open set in X.
When are intersections in a metric space open?
Let U and V be open sets in the metric space (X, d). Then U ⋂ V is an open set. Furthermore, the intersection of any finite family of open sets is open. (Note this does not necessarily hold for infinite families.)
When are unions in a metric space open?
If Uᵢ, i ∈ I is any family of open sets in X, then ⋃ Uᵢ is open.
iϵI
When does a sequence converge in a metric space?
Let (aₙ) be a sequence in a metric space (X, d) and let a ∈ X. Then aₙ ⟶ a as n ⟶ ∞ if and only if for every open set U containing a, there is some N ∈ IN such that aₙ ∈ U for all n > N.
Continuity of a function between metric spaces
Let (X, dᵪ) and (Y, dᵧ) be metric spaces, and let f : X ⟶ Y. Then f is continuous if and only if, for every open set U in Y, the set
f⁻¹(x) = {x ∈ X : f(x) ∈ U} is an open set in X.
Lipschitz & topologically equivalent
If d₁ and d₂ are Lipschitz equivaletn metrics on X then they are topologically equivalent.
Properties of a basis of a topology
If ℬ is a basis for a topology T on X, then
(B1) For each x ∈ X, there is some B ∈ ℬ with x ∈ ℬ
(B2) If x ∈ B₁ and x ∈ B₂ with B₁, B₂ ∈ ℬ then there exists B₃ ∈ ℬ with x ∈ B₃ ⊆ B₁ ⋂ B₂
Conversely, let ℬ be a collection of subsets of a non-empty set X. If ℬ satisfies (B1), (B2) then there is a unique topology T on X such that B is a basis for T.
De Morgan’s Laws
⋃ (X \ Uᵢ) = X \ ( ⋂ Uᵢ)
iϵI iϵI
⋂ (X \ Uᵢ) = X \ ( ⋃ Uᵢ)
iϵI iϵI
Intersections and unions of closed sets in a topological space
In a topological space,
(i) An arbitrary intersection of closed sets is closed
(ii) A finite union of closed sets is closed
Composition of continuous maps between topological spaces
If f : X ⟶ Y and g : Y ⟶ Z are continuous maps between topological spaces, then g ⚬ f : X ⟶ Z is continuous.
Properties of A°
(i) A° is the (unique) largest open subset contained in A, i.e. A° is an open set, A° ⊆ A, and if U is open and U ⊆ A then U ⊆ A°.
(ii) For x ∈ X we have x ∈ A° ⟺ there exists an open set U with x ∈ U ⊆ A.
(iii) A° = A ⟺ A is open.
Properties of Aࠡ
(i) Aࠡ is the (unique) smallest closed subset containing A, i.e. Aࠡ is a closed set, A ⊆ Aࠡ, and if C is closed and A ⊆ C then Aࠡ ⊆ C.
(ii) For x ∈ X we have x ∈ Aࠡ ⟺ there is no open set U with x ∈ U and U ⋂ A = ∅.
(iii) Aࠡ = A ⟺ A is closed.
Application of closures in convergent sequences
Let X be any topological space and let S be a subset of X. Let (aₙ) be a sequence in X with aₙ ∈ S for all n. If aₙ converges to some point a ∈ X then a is in the closure of S.
Convergence of sequences in Hausdorff space
In a Hausdorff space, any sequence can converge to at most one point
Function between Hausdorff spaces
If f : X ⟶ Y is injective and continuous, and Y is Hausdorff, so is X.
Hausdorff and homeomorphic
If X and Y are homeomorphic then X is Hausdorff if and only if Y is Hausdorff.
Properties of the inclusion map
Let A be a non-empty subset of a topological space X (equipped with its subspace topology) and let i : A ⟶ X be the inclusion map. Then
(i) i is continuous
(ii) For any topological space Z and any function g : Z ⟶ A, g is continuous ⟺ i ⚬ g : Z ⟶ X is continuous
(iii) The subspace topology on A is the only topology for which property (ii) holds for all functions g.
Product topology on X x Y where X = Y = R
Let X = Y = R with its usual topology. Then the product topology on R² agrees with the usual topology (given by the Euclidean metric) on R².
When are the projection functions continuous?
Let X, Y be topological spaces, and let pᵪ : X x Y ⟶ X and
pᵧ : X x Y ⟶ Y be the projection functions:
pᵪ( (x, y) ) = x pᵧ( (x, y) ) = y
For any topological space Z and any function f : Z ⟶ X x Y,
f is continuous ⟺ pᵪ ⚬ f and pᵧ ⚬ f are continuous
In particular, pᵪ and pᵧ are continuous
When is f x g continuous?
Let f : X ⟶ X’ and g : Y ⟶ Y’ be continuous functions, and define f x g : X x Y ⟶ X’ x Y’ by
(f x g)(x, y) = (f(x), g(y))
Then f x g is continuous.
Diagonal map
For any topological space X, the diagonal map 𝚫 : X ⟶ X x X, 𝚫(x) = (x, x), is continuous.
Arithmetic of continuous functions
For continuous functions f, g : X ⟶ R, the functions f + g, f - g, fg, etc are continuous.
When is a topological space compact
Let X be a topological space. X is compact if for any family of open sets Uᵢ, i ∈ I with
X = ⋃ Uᵢ
iϵI
we have
n
X = ⋃ Uᵢⱼ
j=1
for some finite subset {i₁, ..., iₙ} of I.Bounded vs compact subsets of a metric space
Let (X, d) be a metric space. Then any compact subset A of X is bounded, i.e. given x ∈ X, there is a real number R such that dₓ(a, x) < R for all a ∈ A.
Compact vs closed and bounded
A compact subset of a metric space is closed and bounded. In particular, any compact subset of R is closed and bounded.
Heine-Borel Theorem
Let a, b be real numbers with a < b. Then the closed, bounded interval [a, b] is compact.
Compact subsubsets
Let C be a compact subset in a topological space X and let A be a closed subset of X with A ⊆ C. Then A is compact.
Compact subsets of R
A subset of R is compact if and only if it is closed and bounded.
Properties of the Middle-Third Cantor Set
(i) Each point x in the Middle-Third Cantor Set A has a unique ternary expansion such that cⱼ ≄ 1 for all j.
(ii) A is uncountable infinite.
(iii) The interior A° of A (considered as a subset of R) is the empty set.
Continuous image of a compact space
The continuous image of a compact space is compact, i.e. if
f : X ⟶ Y is a continuous function between topological spaces, and X is compact, then the subset f(X) of Y is also compact.
When is f : X ⟶ R bounded?
If X is any compact topological space and f : X ⟶ R is any continuous function, then f is bounded and attains its bounds, i.e. the subset f(X) of R is bounded and there are elements
x₁, x₂ ∈ X where f attains its maximum and minimum:
f(x₁) = max{f(x) : x ∈ X} and f(x₂) = min{f(x) : x ∈ X}
When is f a homeomorphism?
If f : X ⟶ Y is a continuous bijection with X compact and Y Hausdorff then f is a homeomorphism.
Product of compact topological spaces
Let X and Y be compact topological spaces. Then their product X x Y is compact.
Compact subsets of Rⁿ
For any n ≥ 1, a subset of Rⁿ is compact if and only if it closed and bounded.
Connected intervals
Any interval (a, b) with a < b is connected
Equivalent definitions of connected
For a topological space X, the following are equivalent:
(i) X is connected
(ii) There is no partition of X
(iii) The only subsets of X which are both open and closed are ∅ and X.
Continuous function on a connected space
Let f : X ⟶ Y be a continuous function between topological spaces. If X is connected, so is f(X).
Union of connected sets
Let X be a topological space, let x ∈ X, and Vᵢ, i ∈ I ≄ ∅ be a family of connected sets with x ∈ Vᵢ for each i. Then
⋃ Vᵢ
iϵI
is connected.
Intersection of connected components
For any x, y ∈ X, either Cₓ = Cᵧ or Cₓ ⋂ Cᵧ = ∅
Closure of a connected set
If A is a connected subset of a topological space X, then its closure Aࠡ is also connected.
Are connected components open/closed?
Connected components are closed. If there are finitely many of them, they are also open.
Does connected imply path connected in Rⁿ?
A connected open subset U of Rⁿ is path connected.
Does path connected imply connected?
Any path connected topological space is connected.
Relationship between Cauchy and convergent sequences
A sequence in R converges (to an element of R) if and only if it is a Cauchy sequence.
Relationship between complete and closed
In a complete metric space, a subspace is complete if and only if it is closed.
Product of complete metric spaces
The product of 2 complete metric spaces is complete.
Banach’s Fixed Point Theorem
Let (X, d) be a complete metric space and let f : X ⟶ X be a contraction. Then f has a unique fixed point, i.e. there is a unique p ∈ X with f(p) = p.
Union of two null sets
If A and B are null sets, so is A ⋃ B.
Union of countably many null sets
The union of countably many null sets is a null set.
Properties of m*
(i) A is a null set if and only if m(A) = 0
(ii) If A ⊆ B then m(A) ≤ m(B): this is clear since if B ⊆ ⋃ Iₙ (n = 1 to ∞) for open intervals Iₙ then also A ⊆ ⋃ Iₙ (n = 1 to ∞)
(iii) m is translation-invariant: if c ∈ R and
A + c = {a + c : a ∈ A},
we have m(A + c) = m(A) since
∞ ∞
A ⊆ ⋃ Iₙ ⟺ A + c ⊆ ⋃ (Iₙ + c)
n=1 n=1
and m(Iₙ + c) = Iₙ.
Countable subbadditivity of m*
m* satisfies the countable subadditivity property:
∞ ∞
m* ( ⋃ Aⱼ ) ≤ ∑ m*(Aⱼ)
j=1 j=1
When is m*(I) = m(I)?
Let I ⊆ R be any interval (not necessarily open nor bounded). Then m*(I) = m(I).
When is m*(I) = m(I)?
Let I ⊆ R be any interval (not necessarily open nor bounded). Then m*(I) = m(I).
Showing that E ∈ ℳ
By the subadditivity of m, we always have
m(A) = m( (A ⋂ E) ⋃ (A ⋂ Eᶜ) )
≤ m(A ⋂ E) + m(A ⋂ Eᶜ)
so to show that E ∈ ℳ, it is enough to show that
m(A ⋂ E) + m(A ⋂ Eᶜ) ≤ m(A).
Properties of ℳ
(i) If E is a null set then E ∈ ℳ. Indeed,
m(A ∩ E) ≤ m(E) = 0
and similarly for m*(A ∩ Eᶜ). In particular, ∅ ∈ ℳ.
(ii) If E ∈ ℳ then Eᶜ ∈ ℳ (since (Eᶜ)ᶜ = E).
(iii) ℳ is translation-invariant:
if E ∈ ℳ then E + t = {x + t : x ∈ E} ∈ ℳ.
Countable unions and intersections on ℳ
ℳ admits countable unions and intersections: if E₁, E₂, … ∈ ℳ then
∞ ∞
⋃ Eₙ ∈ ℳ and ⋂ Eₙ ∈ ℳ
n=1 n=1
Moreover, m* is additive on countable disjoint unions of sets in ℳ: if Eᵢ ⋂ Eⱼ = ∅ for i ≄ j, then
∞ ∞
m* ( ⋃ Eₙ) = ∑ m*(Eₙ)
n=1 n=1
Intervals in ℳ
Every interval is in ℳ
Subsets of R in ℳ
Every open subset of R is in ℳ, and every closed subset of R is in ℳ.
Intersection of 𝜎-algebras
Let {ℬᵢ : i ∈ I} be any set of 𝜎-algebras on X. Then their intersection
ℬ = ⋂ ℬᵢ
i ϵ I
is also a 𝜎-algebra on X.
Lebesgue measurable vs Borel measurable
There are functions f : R ⟶ R which are Lebesgue measurable but not Borel measurable (e.g. indicator function of a set which is Lebesgue measurable but not Borel measurable).
