METRIC SPACES

Question 1

f: (X,dx) -> (Y,dy) continuous

Answer 1

⟺ f(A_) ⊆ f(A)_ , ∀A ⊆X
⟺ f^(-1)(V) open in X whenever V open in Y
⟺ f^(-1)(V) closed in X whenever V closed in Y

Question 2

f: (X,dx) -> (Y,dy) continuous at a

Answer 2

ε>0, ∃δ>0:
dy(f(x),f(a)) < ε ⟹ dx(x,a)<δ

Question 3

A⊆(X,d) bounded

Answer 3

∃x∈X , ∃k∈R:
d(a,x)<=k, ∀a∈A

Question 4

diameter of non-empty bounded A⊆(X,d)

Answer 4

sup{d(x,y): x,y∈A}

Question 5

open ball in X centred at a with radius r

Answer 5

Br(a) = {x∈X: d(x,a)<r}

Question 6

A⊆X open in X

Answer 6

∀a∈A, ∃ε>0: Bε(a)⊆A

Question 7

x∈X is limit point of A⊆X

Answer 7

ε>0, (Bε(x)\{x})∩A ≠ ∅

Question 8

interior of A⊆X

Answer 8

A∘ := {a∈A: Bε(a)⊆A for some ε>0}

Question 9

(xn) converges to a point x∈X

Answer 9

ε>0, ∃N: xn∈Bε(x) whenever n>=N

Question 10

(xn) Cauchy

Answer 10

ε>0, ∃N: d(xm,xn) < ε whenever m,n>=N

Question 11

d1, d2 on X “Lipschitz equivalent”
⟹

Answer 11

∃h,k positive constants:
∀x,y∈X, hd2(x,y)<=d1(x,y)<=kd2(x,y)

Question 12

d1,d2 “topologically equivalent”

Answer 12

A⊆X d1-open in X
⟺
A⊆X d2-open in X

Question 13

isometry
f:(X,dx)->(Y,dy)

Answer 13

f bijective such that
dy(f(x1),f(x2)) = dx(x1,x2), ∀x1,x2∈X

Question 14

A⊆X dense in X

Answer 14

A_ = X

Question 15

x point of closure of A⊆(X,d)

Answer 15

ε>0, Bε(x)∩A ≠ ∅

Question 16

f:X->Y “uniformly continuous” on X

Answer 16

ε>0, ∃δ>0:
dy(f(x),f(x’)) < ε ⟹ dx(x,x’)<δ
∀x,x’∈X

X compact, f continuous

Question 17

Lebesque number

Answer 17

• C is cover for A.
• lebesque number for C is a real ε>o:
  B↓(ε) (a) contained in single set of C for ∀a∈A.

Question 18

ε-net for X

Answer 18

N⊆X : {B↓(ε) (x): x∈N} covers X, some ε>0

Question 19

X “complete”

Answer 19

if every Cauchy sequence in X converges (to a point of X)

Question 20

f:X->X “contraction”

Answer 20

d(f(x),f(y))<=Kd(x,y), ∀x,y∈X.
for some constant K<1

Question 21

Banach’s Fixed Point Theorem

Answer 21

f:X->X contraction, metric space X complete ⟹ f has a unique fixed point p in X

Question 22

x∈X is in boundary of A⊆X

Answer 22

∀ε>0, A∩Bε(x) and (X\A)∩Bε(x) non-empty

Question 23

Cauchy (xn) converges to a point x∈X

Answer 23

if (xn) has subsequence converging to x

Question 24

X “sequentially compact”

Answer 24

every sequence in X has at least one convergent subsequence in X
⟺
X compact

Question 25

X “sequentially compact”
⟹

Answer 25

• any open cover for X has lebesque number
• X has finite ε-net

Question 26

X “regular” / “normal”

Answer 26

given V⊆X closed and x∈X\V,
∃U,U’⊆X open disjoint:
V⊆U and x∈U’

Question 27

f : (X,Tx) -> (Y,Ty) continuous

Answer 27

if V∈Ty ⟹ f^(-1)(V)∈Tx
⟺
f(A_)⊆f(A)_ , ∀A⊆X
⟺
f^(-1)(V) open in X whenever V open in Y
⟺
f^(-1)(V) closed in X whenever V closed in Y

Question 28

f : (X,Tx) -> (Y,Ty) continuous at a point x∈X

Answer 28

if, given any V∈Ty : f(x)∈V,
there is some U∈Tx: x∈U and f(U)⊆V

Question 29

basis for T

Answer 29

subfamily B⊆T :
every set in T is a union of sets from B.

Question 30

X “connected”

Answer 30

X admits no partition

Question 31

partition {A,B} of X

Answer 31

A,B ⊆ X open, non-empty:
X = A∪B, A∩B=∅

Question 32

A path f “joins” x and y in X

Answer 32

a continuous map f:[0,1]->X such that f(0) = x and f(1)=y

Question 33

X “path-connected”

Answer 33

if any two points of X can be joined by a path in X
⟹ X connected

Question 34

X⊆R^n open connected ⟹

Answer 34

X path-connected

Question 35

homeomorphism

Answer 35

bijective map f :
f and f^(-1) continuous

Question 36

A⊆X “compact”

Answer 36

if every open cover for A has a finite sub cover

Question 37

A⊆X “relatively compact”

Answer 37

if A_ compact in X

Question 38

Heine-Borel theorem

Answer 38

any closed bounded subset of R^n is compact

Question 39

connected A⊆X
A ⊆B ⊆A_ ⟹

Answer 39

B connected

Question 40

f has IVP

Answer 40

f(a)<x<f(b)
∃c ∈[a,b] : f(c) = x

Question 41

f:D->R satisfies Lipschitz condition of order α on D

Answer 41

α,K>0,
|f(x) - f(y)|<= K|x-y|^α, ∀x,y∈D.
k constant

Question 42

Cauchy’s criterion for uniform convergence

Answer 42

(fn) converges uniformly on D
⟺
(fn) uniformly Cauchy on D

Question 43

(fn) “uniformly Cauchy” on D⊆R

Answer 43

if given ε>0, ∃ N∈N:
|fm(x) - fn(x)|<ε , ∀m,n>=N and ∀x∈D

Question 44

(fn) converges uniformly to f on D

Answer 44

ε>0, ∃N∈N:
|fn(x) - f(x)|<ε, ∀n>=N, ∀x∈D
⟺
sup|fn(x)-f(x)| exists for large n and limit is 0 as n -> ∞

Question 45

(fn) converges pointwise on D

Answer 45

(fn(x)) converges to f(x), ∀x∈D

Question 46

Bolzano-Weierstrass theorem

Answer 46

every bounded sequence of real numbers has at least one convergent subsequence

Question 47

Sup(X)

Answer 47

the upper bound s is supremum if ∀ε>0, ∃x∈X : s-ε < x

Question 48

axiom of completeness

Answer 48

any non-empty set of real numbers that is bounded above has a least upper bound