Theorems

Question 1

Intermediate Value Theorem

Answer 1

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.

Question 2

Bounds of a continuous function [a, b] ⟶ R.

Answer 2

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].

Question 3

When is B_𝜀(a) open?

Answer 3

For any a ∈ X and any 𝜀 > 0, the set B_𝜀(a) is an open set in X.

Question 4

When are intersections in a metric space open?

Answer 4

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.)

Question 5

When are unions in a metric space open?

Answer 5

If Uᵢ, i ∈ I is any family of open sets in X, then ⋃ Uᵢ is open.
iϵI

Question 6

When does a sequence converge in a metric space?

Answer 6

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.

Question 7

Continuity of a function between metric spaces

Answer 7

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.

Question 8

Lipschitz & topologically equivalent

Answer 8

If d₁ and d₂ are Lipschitz equivaletn metrics on X then they are topologically equivalent.

Question 9

Properties of a basis of a topology

Answer 9

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.

Question 10

De Morgan’s Laws

Answer 10

⋃ (X \ Uᵢ) = X \ ( ⋂ Uᵢ)
iϵI iϵI

⋂ (X \ Uᵢ) = X \ ( ⋃ Uᵢ)
iϵI iϵI

Question 11

Intersections and unions of closed sets in a topological space

Answer 11

In a topological space,

(i) An arbitrary intersection of closed sets is closed
(ii) A finite union of closed sets is closed

Question 12

Composition of continuous maps between topological spaces

Answer 12

If f : X ⟶ Y and g : Y ⟶ Z are continuous maps between topological spaces, then g ⚬ f : X ⟶ Z is continuous.

Question 13

Properties of A°

Answer 13

(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.

Question 14

Properties of Aࠡ

Answer 14

(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.

Question 15

Application of closures in convergent sequences

Answer 15

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.

Question 16

Convergence of sequences in Hausdorff space

Answer 16

In a Hausdorff space, any sequence can converge to at most one point

Question 17

Function between Hausdorff spaces

Answer 17

If f : X ⟶ Y is injective and continuous, and Y is Hausdorff, so is X.

Question 18

Hausdorff and homeomorphic

Answer 18

If X and Y are homeomorphic then X is Hausdorff if and only if Y is Hausdorff.

Question 19

Properties of the inclusion map

Answer 19

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.

Question 20

Product topology on X x Y where X = Y = R

Answer 20

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.

Question 21

When are the projection functions continuous?

Answer 21

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

Question 22

When is f x g continuous?

Answer 22

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.

Question 23

Diagonal map

Answer 23

For any topological space X, the diagonal map 𝚫 : X ⟶ X x X, 𝚫(x) = (x, x), is continuous.

Question 24

Arithmetic of continuous functions

Answer 24

For continuous functions f, g : X ⟶ R, the functions f + g, f - g, fg, etc are continuous.

Question 25

When is a topological space compact

Answer 25

``` 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. ```

Question 26

Bounded vs compact subsets of a metric space

Answer 26

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.

Question 27

Compact vs closed and bounded

Answer 27

A compact subset of a metric space is closed and bounded. In particular, any compact subset of R is closed and bounded.

Question 28

Heine-Borel Theorem

Answer 28

Let a, b be real numbers with a < b. Then the closed, bounded interval [a, b] is compact.

Question 29

Compact subsubsets

Answer 29

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.

Question 30

Compact subsets of R

Answer 30

A subset of R is compact if and only if it is closed and bounded.

Question 31

Properties of the Middle-Third Cantor Set

Answer 31

(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.

Question 32

Continuous image of a compact space

Answer 32

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.

Question 33

When is f : X ⟶ R bounded?

Answer 33

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}

Question 34

When is f a homeomorphism?

Answer 34

If f : X ⟶ Y is a continuous bijection with X compact and Y Hausdorff then f is a homeomorphism.

Question 35

Product of compact topological spaces

Answer 35

Let X and Y be compact topological spaces. Then their product X x Y is compact.

Question 36

Compact subsets of Rⁿ

Answer 36

For any n ≥ 1, a subset of Rⁿ is compact if and only if it closed and bounded.

Question 37

Connected intervals

Answer 37

Any interval (a, b) with a < b is connected

Question 38

Equivalent definitions of connected

Answer 38

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.

Question 39

Continuous function on a connected space

Answer 39

Let f : X ⟶ Y be a continuous function between topological spaces. If X is connected, so is f(X).

Question 40

Union of connected sets

Answer 40

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.

Question 41

Intersection of connected components

Answer 41

For any x, y ∈ X, either Cₓ = Cᵧ or Cₓ ⋂ Cᵧ = ∅

Question 42

Closure of a connected set

Answer 42

If A is a connected subset of a topological space X, then its closure Aࠡ is also connected.

Question 43

Are connected components open/closed?

Answer 43

Connected components are closed. If there are finitely many of them, they are also open.

Question 44

Does connected imply path connected in Rⁿ?

Answer 44

A connected open subset U of Rⁿ is path connected.

Question 45

Does path connected imply connected?

Answer 45

Any path connected topological space is connected.

Question 46

Relationship between Cauchy and convergent sequences

Answer 46

A sequence in R converges (to an element of R) if and only if it is a Cauchy sequence.

Question 47

Relationship between complete and closed

Answer 47

In a complete metric space, a subspace is complete if and only if it is closed.

Question 48

Product of complete metric spaces

Answer 48

The product of 2 complete metric spaces is complete.

Question 49

Banach's Fixed Point Theorem

Answer 49

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.

Question 50

Union of two null sets

Answer 50

If A and B are null sets, so is A ⋃ B.

Question 51

Union of countably many null sets

Answer 51

The union of countably many null sets is a null set.

Question 52

Properties of m*

Answer 52

(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.

Question 53

Countable subbadditivity of m*

Answer 53

m* satisfies the countable subadditivity property: ∞ ∞ m* ( ⋃ Aⱼ ) ≤ ∑ m*(Aⱼ) j=1 j=1

Question 54

When is m*(I) = m(I)?

Answer 54

Let I ⊆ R be any interval (not necessarily open nor bounded). Then m*(I) = m(I).

Question 55

When is m*(I) = m(I)?

Answer 55

Let I ⊆ R be any interval (not necessarily open nor bounded). Then m*(I) = m(I).

Question 56

Showing that E ∈ ℳ

Answer 56

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).

Question 57

Properties of ℳ

Answer 57

(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.

Question 58

Countable unions and intersections on ℳ

Answer 58

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

Question 59

Intervals in ℳ

Answer 59

Every interval is in ℳ

Question 60

Subsets of R in ℳ

Answer 60

Every open subset of R is in ℳ, and every closed subset of R is in ℳ.

Question 61

Intersection of 𝜎-algebras

Answer 61

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.

Question 62

Lebesgue measurable vs Borel measurable

Answer 62

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).