Definitions (Billy Cole)

Question 1

define a metric space (X,d)

Answer 1

(X,d) consists of a non empty set X and a non-negative real valued metric d:XxX → ℝ* which satisfies the axioms.

1. d(x,y)=0 x=y Ɐx,y ∈ X
2. d(x,y)=d(y,x) Ɐx,y ∈ X
3. d(x,z) ≤ d(x,y)+d(y,z) Ɐx,y,z ∈ X

Question 2

define a Subspace

Answer 2

Given any subset W⊆X, the restriction of d to W determines the subspace (W, d:= d_w) of (X,d).

Question 3

Define an open ball

Answer 3

For any metric space (X,d), the open ball of radius r>0 around point x∈X is the subspace,
B_r(x) = {y : d(y,x) < r} .

Question 4

Define a closed ball

Answer 4

For any metric space (X,d), the open ball of radius r>0 around point x∈X is the subspace,
Ɓ_r(x) = {y : d(y,x) ≤ r} .

Question 5

Define an Euclidean n-space (ℝ^n,d_2)

Answer 5

Euclidean n-space (ℝ^n,d_2) consists of all real n-dimensional vectors x=(x_1,…,x_n), equipped with the Euclidean metric
d_2(x,y) = [(x_1 - y_1)²+ … +(x_n - y_n)²]^½
where the positive square root is understood.

Question 6

what is the Euclidean metric

Answer 6

d_2(x,y) = [(x_1 - y_1)²+ … +(x_n - y_n)²]^½

Question 7

What is the open ball of the Euclidean metric?

Answer 7

B_r(x) = {y ∈ℝ^n : (⅀(y_i - x_i)^2)^½ < r} .

Question 8

What is the closed ball of the Euclidean metric?

Answer 8

Ɓ_r(x) = {y ∈ℝ^n : (⅀(y_i - x_i)^2)^½ ≤ r}

Question 9

what is the taxicab metric?

Answer 9

d_1(x,y) = |x_1 - y_1|+ … + |x_n - y_n|

Question 10

what is the discrete metric?

Answer 10

for any non empty set X,
d(x,y) = {0 if x=y
{1 otherwise

Question 11

what is the open Ball of the discrete metric?

Answer 11

B_r(x) = { {x} if r ≤ 1

{ X if r > 1

Question 12

what is the closed Ball of the discrete metric?

Answer 12

Ɓ_r(x) = { {x} if r < 1

{ X if r ≥ 1

Question 13

define isometry

Answer 13

For any two metric spaces (X, d_X) and (Y, d_Y ), a bijection
f:X→Y is an isometry whenever d_X(x,y) = d_Y(f(x), f(y)) for all Ɐx,y ∈ X

Question 14

define the standard metric d_ℂ

Answer 14

d_ℂ on the complex numbers ℂ is given by
d_ℂ(z,z’) = |z-z’|
where |z|= r

Question 15

what are the balls of the standard metric, d_ℂ, like?

Answer 15

the balls in ℂ are genuine discs of radius r.

Question 16

define a Graph. Γ=(V,E)

Answer 16

Γ=(V,E) consists of a set V of vertices, and a set E of edges; each edge vw ∈E may be interpreted as joining two vertices v,w ∈V.

Question 17

define a path, in a graph Γ

Answer 17

a path in Γ from u to w is a finite sequence of edges

π(u,w) = (uv_1, v_1v_2, ….. , v_(n-2)v_(n-1), v_(n-1)w).

Question 18

whats the edge metric e on the vertex set V of a path connected graph is defined by ?

Answer 18

e(u,w) = min l(π(u,w))

_π(u,w)

Question 19

what is the open ball in the edge metric

Answer 19

B_r(u) = {w∈V : π(u,w) exists with <r edges}

Question 20

what is the closed ball in the edge metric

Answer 20

Ɓ_r(u) = {w∈V : π(u,w) exists with ≤r edges}.

Question 21

whats the d_min on X

Answer 21

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

{½^n if n=min{m:x_m≠y_m}

Question 22

Define bounded

in terms of a function

Answer 22

A real-valued function f on a closed interval [a,b]⊂ℝ is bounded whenever there exists a constant K such that |f(x)|≤K for every x∈[a,b].

but the converse it false

Question 23

whats the sup metric

Answer 23

d_sup(f,g) = sup |f(x)-g(x)|

x∈[a,b]

Question 24

what is ℬ[a,b] ?

Answer 24

It is the metric space (X,d_sup) for any interval [a,b]⊆ℝ.

Question 25

what is ℒ_1[a,b] ?

Answer 25

The metric space (Y,d_1), where d-1 is the metric that measures the area between functions. d_1(f,g) = ∫ |f(t)-g(t)| dt

Question 26

what is ℒ_2[a,b] ?

Answer 26

The metric space (Y,d_2), where d_2 equals d_2(f,g) = ( ∫ ( f(t)-g(t) )² dt)^½

Question 27

what is the interval metric, d_H on X.

Answer 27

X is the set of all closed intervals [a,b] in the euclidean line. d_H([a,b],[r,s]) = max{ | r - a |, | s - b | }.

Question 28

what is the l_1 metric d_1

Answer 28

d_1( (a_i),(b_i) ) = ⅀|a_i - b_i|

Question 29

in terms of d_1 what are the balls of the l_1 metric

Answer 29

B_r((a_i)) = {(b_i) : ⅀|a_i - b_i| < r } and Ɓ_r((a_i)) = {(b_i) : ⅀|a_i - b_i| ≤ r }

Question 30

define the cartesian product of two metric spaces (X,d) and (X',d').

Answer 30

the cartesian product of two metric spaces (X,d) and (X',d') is the set XxX' equipped with one the metrics 1. d_a((x,x'),(y,y')) = d(x,y) + d'(x',y') 2. d_b((x,x'),(y,y')) = (d(x,y)² + d'(x',y')²)^½ 3. d_c((x,x'),(y,y')) = max{d(x,y), d'(x',y')}

Question 31

define Lipschitz equivalent. | in terms of metrics

Answer 31

two metrics d and e on X are lipschitz equivalent whenever there exists positive constants h,k∈ℝ such that; he(x,y) ≤ d(x,y) ≤ ke(x,y) for every x,y∈ X

Question 32

Define an interior point u

Answer 32

An Interior point u∈U is one for which there exists ε>0 such that B_ε(u)⊆U.

Question 33

Define the Interior of U

Answer 33

The Interior of U is the subset U°⊆U of all interior points.

Question 34

define when U is open in X

Answer 34

when U°=U.

Question 35

Define closure point.

Answer 35

a point x∈X is a closure point of U⊆X if B_ε(u)⋂U is non-empty for every ε>0.

Question 36

Define closure.

Answer 36

The Closure of U is the superset Ū⊇U of all closure points.

Question 37

Define closed

Answer 37

U is closed in X if Ū=U.

Question 38

Define a Partially open ball in X

Answer 38

a partially open ball in X is a set P_r(x) = B_r(x)⋃P, where P and Q are non-empty disjoint subsets satisfying P⋃Q={p:d(x,p)=r}.

Question 39

Define a sequence

Answer 39

For any metric space X = (X, d), a sequence in X is a function s: ℕ → X

Question 40

Define converges

Answer 40

``` A sequence (x_n) converges to the point x ∈ X whenever ∀ε > 0, ∃N ∈ ℕ such that n ≥ N ⇒ d(x, x_n) ```

Question 41

Define convergence in terms of open balls

Answer 41

x_n → x whenever | ∀ε > 0, ∃N∈ℕ such that n ≥ N ⇒ x_n ∈ B(x).

Question 42

Define a Cauchy sequence

Answer 42

In any metric space (X; d), a Cauchy sequence (x_n) satisfies ∀ε > 0, ∃N ∈ℕ such that m,n ≥N ⇒ d(x_m, x_n)

Question 43

Define Dense

Answer 43

A subset Y is dense in (X, d) whenever Ȳ = X.

Question 44

Define Bounded | in terms of a metric space

Answer 44

A subset A of a metric space (X,d) is bounded whenever | there exists x_0 ∈X and M∈R such that d(x, x_0) ≤M for every x∈A.

Question 45

Define the diameter

Answer 45

diam(A) of a bounded non-empty subset A⊆X is the real number | sup {d(x,y) : x,y∈A}

Question 46

Define a Boundary Point

Answer 46

A boundary point x∈X of A is one for which every open | ball B_ε(x) meets both A and X\A.

Question 47

Define a the Boundary ∂A

Answer 47

The boundary ∂A of A is the set of all boundary points.

Question 48

The boundary of any subset A is ?

Answer 48

the set Ᾱ\A°

Question 49

Define pointwise convergence

Answer 49

A sequence (f_n: n≥1) of such functions converges pointwise to f on D whenever the sequence of real numbers ( f_n(x) ) converges to f(x) in ℝ for every x∈D.

Question 50

Define Uniform convergence

Answer 50

A sequence (f_n: n≥1) of functions converges uniformly to f on D whenever ∀ε>0, ∃N(ε) ∈ℕ such that n≥N(ε) ⇒ | f_n(x) - f(x) |ε for every x∈D.

Question 51

f_n →f uniformly on D ⇒ ?

Answer 51

f_n→ f pointwise on D

Question 52

Define continuous at x_0 in X | wol lol lor

Answer 52

Given metric spaces (X,d_x) and (Y,d_y), a function f: X→Y is continuous at x_0 in X whenever ∀ε>0; ∃δ>0 such that d_x(x,x_0)

Question 53

Define a map

Answer 53

a continuous function.

Question 54

define when f is continuous

Answer 54

f is continuous at every x_0∈X, then f is continuous.

Question 55

define when f is continuous at x_0 in X, in terms of open balls

Answer 55

Given metric spaces (X,d_x) and (Y,d_y), a function f: X→Y is continuous at x_0 in X whenever ∀ε>0, ∃δ>0 such that x∈B_δ(x_0) ⇒ f(x) ∈B_δ(f(x_0))

Question 56

Define the Inverse image

Answer 56

For any function f: X→Y and any subset U⊆Y , the inverse image f^-1(U) is the subset {x : f(x)∈U} ⊆X.

Question 57

Define Lipschitz Equivalence | in terms of metric spaces

Answer 57

For any two metric spaces (X,d_x) and (Y,d_y), a bijection f: X→Y is a Lipschitz equivalence whenever there exist positive constants h,k∈ℝ such that hd_y (f(w), f(x)) ≤ d_x(w,x) ≤ kd_y (f(w), f(x)) for all w,x∈X.

Question 58

Define a homeomorphism

Answer 58

A bijection f: X→Y is a homeomorphism whenever f and | f^-1 are both continuous.

Question 59

Define topologically equivalent

Answer 59

Two metrics d and e on a set X are topologically equivalent whenever the identity function 1_x is a homeomorphism.

Question 60

Define Path connected

Answer 60

A metric space X is path connected if every two points x_0, x_1 ∈X admit a continuous function σ: [0,1]→X such that σ(0) = x_0 and σ(1) = x_1. Then σ is a path from x_0 to x_1 in X

Question 61

Define a Covering of A

Answer 61

is a collection of sets Ʋ = {U_i : i ∈ I} for which | A ⊆ ⋃U_i

Question 62

Define a Subcovering of Ʋ

Answer 62

a subcovering of U is a subcollection {U_i : i∈J} which also covers A, for some J⊆I.

Question 63

Define a open covering

Answer 63

If every U_i is open, then U is an open covering of A

Question 64

Define Compact

Answer 64

A subset A⊆X is compact if every open covering of A contains a finite subcovering.

Question 65

Define Sequentially compact

Answer 65

A subspace S⊆X is sequentially compact if any infinite sequence (x_n : n≥0) in S has a subsequence (x_r : r≥1) that converges to a point in S

Question 66

Define the Cantor set K

Answer 66

The Cantor set K⊂ℝ is defined by the following inductive procedure. Let K_0= [0,1], and define K_1⊂K_0 by deleting its open middle third. so K_1 = [0, ⅓]⋃[ ⅔, 1] is the union of two closed intervals, each of length ⅓. Let K_n be the union of 2^n closed intervals, each of length ⅓^n, and define K_n+1 by deleting the open middle third of each; so K_n+1 is the union of 2^n+1 closed intervals, each of length ⅓^(n+1). then K= K_1 ⋂ ... ⋂ K_n ⋂...

Question 67

The Cantor set is ?

Answer 67

uncountable closed compact

Question 68

Define complete

Answer 68

A metric space (X, d) is complete if every Cauchy sequence tends to a limit in X.

Question 69

Define a contraction

Answer 69

Given any metric space (X, d), a self-map f : X → X is a | contraction whenever there exists a constant 0

Question 70

Define fixed point

Answer 70

A fixed point of a self-map f : X → X is a point x ∈ X for which f(x) = x