Flashcards in Metric Spaces Deck (47):

1

## Metric

###
For a set M, d: MxM → ℝ is a metric if ∀ x,y,z ∈ M

i) Positivity: d(x,y) ≥ 0, and d(x,y) = 0 ⇔ x = y

ii) Symmetry: d(x,y) = d(y,x)

iii) Triangle Inequality: d(x,y) ≤ d(x,z) + d(z,y)

(M,d) is a metric space.

2

## Norm

###
A function ‖•‖: M → ℝ is a norm if ∀ x,y ∈ M, ∀ λ ∈ ℝ

i) ‖x‖ ≥ 0 and ‖x‖ = 0 ⇔ x = 0,

ii) ‖λx‖ = |λ|⋅‖x‖,

iii) ‖x + y‖ ≤ ‖x‖ + ‖y‖.

(M,‖•‖) is a normed vector space.

3

## Discrete metric

###
For M set, d: M×M → ℝ is a discrete metric if

d(x,y) = {0 if x = y,

1 if x ≠ y

4

## French railway metric

###
For a normed vector space (M, ‖•‖), d: M×M → ℝ is a French railway metric if

d(x,y) = {‖x - y‖ if x,y,0 collinear, ‖x‖ + ‖y‖ otherwise

5

## Open ball centre a, radius r

###
Given a metric space (M,d), a ∈ M, r > 0,

B(a,r) = {x ∈ M: d(x,a) < r} is the open ball with centre a, radius r.

6

## Open set (MS)

### For a metric space (M,d), U ⊂ M is open if ∀ x ∈ U, ∃ ε > 0 s.t. B(x, ε) ⊂ U.

7

## Closed set (MS)

### For a metric space (M, d), F ⊂ M is closed if M\F is open

8

## DeMorgan's laws

###
X\∪(Aᵢ) = ∩(X\Aᵢ)

X\ ∩(Aᵢ) = ∪(X\Aᵢ)

9

## Convergence of a sequence xₙ in metric space M

###
Let (M, d) be a metric space, (xₙ) ⊂ M a sequence.

xₙ is convergent in M if ∃ L ∈ M s.t. ∀ε > 0, ∃ N ≥ 1 s.t. if n ≥ N, d(xₙ, L) < ε.

Call L the limit, write xₙ → L in M as n → ∞

10

## f continuous at a (MS)

###
Let (M₁, d₁), (M₂, d₂) be metric spaces, f: M₁ → M₂, a ∈ M₁.

f is cts at a if ∀ ε > 0, ∃ δ > 0 s.t. d₁(x, a) < δ ⇒ d₂(f(x), f(a)) < ε.

11

## f continuous (MS)

### f: M₁ → M₂ is cts if f is cts at a, ∀ a ∈ M₁.

12

## f Lipschitz

### A function f: M₁ → M₂ is Lipschitz if ∃ L > 0 s.t. d₂(f(x), f(y)) ≤ L⋅d₁(x, y) ∀ x,y ∈ M₁.

13

## Inverse image of U under f

### Let (M₁, d₁) and (M₂, d₂) be metric spaces, U ⊂ M₂, f: M₁ → M₂. The inverse image of U under f is f⁻¹(U) = {x ∈ M₁ : f(x) ∈ U}.

14

## Homeomorphism (MS)

###
Let M₁, M₂ be metric spaces, f: M₁ → M₂.

f is a homeomorphism if

i) f is a bijection,

ii) f is cts from M₁ to M₂,

iii) f⁻¹ is cts from M₂ to M₁.

15

## Homeomorphic (MS)

### if ∃ f: M₁ → M₂ and f is a homeomorphism, then M₁ and M₂ are homeomorphic.

16

## Topological property

### P is a topological property of metric spaces if it is preserved by homeomorphism - if M₁ is homeomorphic to M₂ then P holds for M₁ ⇔ P holds for M₂.

17

## Bounded (MS)

### For a metric space (M, d), U ⊂ M is bounded if ∃ a ∈ M, r > 0 s.t. M ⊂ B(a, r).

18

## Topology

###
Given a set X, X ≠ ∅, a topology τ on X is a collection of subsets of X (τ ⊂ 𝒫(X)), s.t.

i) X, ∅ ∈ τ

ii) U₁, ..., Uₖ ∈ τ ⇒ U₁ ∩ ... ∩ Uₖ ∈ τ

iii) Uᵢ ∈ τ, i ∈ I ⇒ ∪_(i ∈ I) Uᵢ ∈ τ.

(X, τ) is a topological space.

19

## Open set (TS)

###
Let (X, τ) be a topological space.

U ⊂ X is open if U ∈ τ.

20

## Closed set (TS)

###
Let (X, τ) be a topological space, F ⊂ X.

F is closed if X\F is open.

21

## Neighbourhood (TS)

###
Let (X, τ) be a topological space, x ∈ X, N ⊂ X.

N is a neighbourhood of x if x ∈ N and N is open.

22

## Interior of A ⊂ X in (X, τ)

###
Let (X, τ) be a top space, A ⊂ X.

The interior of A, denoted Å, is Å = {x ∈ A | ∃ nhd N of x, N ⊂ A}

23

## Closure of A ⊂ X in (X, τ)

###
Let (X, τ) be a top space, A ⊂ X.

The closure of A, denoted Ã (A overbar) is Ã = {x ∈ X | every nhd of x intersects A}

24

## Finite complement topology

###
Given a set X, we say (X, τ) is a finite complement topology if τ is such that

τ = {A ⊂ X | A = ∅ or X\A is finite}

25

## Boundary of A ⊂ X in (X, τ)

###
Let (X, τ) be a top space, A ⊂ X.

The boundary of A is

∂A = {x ∈ X : every nhd of X intersects A and X\A}

26

## Facts about Closure, Interior and Boundary of A ⊂ X

###
A open ⇔ A = Å

A closed ⇔ A = Ã

∂A = Ã \ Å

∂A = ∂(X\A)

Ã = A ∪ ∂A

27

## Clos/u/re, I/n/terior

###
(A ∪ B)~ = Ã ∪ B~

(A ∩ B)° = A° ∩ B°

28

## Continuity and Preimages

### f: (M _1, d_1) → (M₂, d₂), is continuous iff f ⁻¹(U) is open for any U ⊂ M₂ open

29

## Hausdorff space (X, τ)

### ∀ x,y ∈ X, we can find nhds of U and V of x and y respectively such that U ∩ V = ∅

30

## Convergence in a metric space and open sets

###
Let (M,d) be a metric space, x_n ∈ M, x ∈ M.

x_n → x ⇔ ∀ U ⊂ M with x ∈ X, ∃ N ≥ 1 s.t. if n ≥ N, x_n ∈ U.

31

## Convergence of a sequence xₙ in a top. space (X, τ)

### xₙ converges to L (xₙ → L as n → ∞) if ∀ nhd U of L, ∃ N ≥ 1 s.t. n ≥ N ⇒ xₙ ∈ U

32

## Cover 𝒪 of X

### 𝒪 ⊂ 𝒫(X) is a cover of X if ∀ x ∈ X, ∃ A ∈ 𝒪 s.t. x ∈ A

33

## Basis ℬ of X

###
ℬ ⊂ 𝒫(X) is a basis of a topology in X if:

i) ℬ is a cover of X

ii) If B₁, B₂ ∈ ℬ and x ∈ B₁ ∩ B₂, then ∃ B₃ ∈ ℬ s.t. x ∈ B₃ ⊂ B₁ ∩ B₂

34

## Topology τ generated by ℬ

###
Given a basis ℬ for a topology on X,

τ = {U ⊂ X : ∀ x ∈ U, ∃ B ∈ ℬ s.t. x ∈ B ⊂ U} is the topology generated by ℬ.

35

## f continuous (TS)

###
f: X₁ → X₂ is cts if f⁻¹(U) is open in X₁, ∀ U open in X₂.

OR f⁻¹(F) closed in X₁ ∀ F closed in X₂

OR f⁻¹(B) open in X₁ ∀ basis elt B ⊂ X₂.

36

## Finer/Coarser topology

### If the identity map is cts from (X, τ₁) to (X, τ₂), then τ₂ ⊂ τ₁, so τ₁ is finer than τ₂, or τ₂ is coarser than τ₁.

37

## Homeomorphism (TS)

###
f: X → Y is a homeomorphism if

i) f is a bijection

ii) f is cts

iii) f⁻¹ is cts

OR f is a bijection and f(U) open in Y ⇔ U open in X

OR f is a bijection and f(F) closed in Y ⇔ F closed in X

38

## Box topology

###
Given X₁, X₂, ... top. spaces, X = X₁×X₂×..., the box topology is τ generated by

ℬ = {∏ᵢ₌₁ ^∞ Uᵢ, Uᵢ open in Xᵢ}.

39

## Product topology

###
Given X₁, X₂, ... top. spaces, X = X₁×X₂×..., the product topology is τ generated by

ℬ = {∏ᵢ₌₁ ^∞ Uᵢ, Uᵢ ⊂ Xᵢ open, ∀ i ≥ k Uᵢ = Xi for some k}

40

## Finite subcover of 𝒪

### A cover 𝒪 has finite subcover if ∃ k ≥ 1, A₁, ..., Aₖ ∈ 𝒪 s.t. {A₁,...,Aₖ} is still a cover (i.e. X = A₁ ∪ ... ∪ Aₖ)

41

## Open cover of (X, τ)

### 𝒪 is an open cover of X if 𝒪 is a cover of X and 𝒪 ⊂ τ

42

## Compact

### A set X is compact if every open cover has a finite subcover

43

## Cover of subspace Y ⊂ X

### 𝒪 ⊂ 𝒫(X) covers Y ⊂ X if Y ⊂ ∪_{A ∈ 𝒪} A

44

## Uniform continuity of f (MS)

###
f: (X₁, d₁) → (X₂, d₂) is uniformly cts if ∀ ε > 0 ∃ δ > 0 s.t. ∀ x, y ∈ X₁,

d₁(x, y) < δ ⇒ d₂(f(x), f(y)) < ε.

45

## Sequentially compact (TS)

### (X, τ) is sequentially compact if ∀ xₙ ∈ X, ∃ nₖ ≥ 1, nₖ₊₁ > nₖ and y ∈ X s.t. x_{nₖ} → y as k → ∞

46

## Separation A,B of X

###
A,B ⊂X are a separation of X if

i) A,B ≠ ∅,

ii) A ∩ B = ∅, A ∪ B = X,

iii) A,B open.

47