METRIC SPACES Flashcards
f: (X,dx) -> (Y,dy) continuous
⟺ 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
f: (X,dx) -> (Y,dy) continuous at a
ε>0, ∃δ>0:
dy(f(x),f(a)) < ε ⟹ dx(x,a)<δ
A⊆(X,d) bounded
∃x∈X , ∃k∈R:
d(a,x)<=k, ∀a∈A
diameter of non-empty bounded A⊆(X,d)
sup{d(x,y): x,y∈A}
open ball in X centred at a with radius r
Br(a) = {x∈X: d(x,a)<r}
A⊆X open in X
∀a∈A, ∃ε>0: Bε(a)⊆A
x∈X is limit point of A⊆X
ε>0, (Bε(x)\{x})∩A ≠ ∅
interior of A⊆X
A∘ := {a∈A: Bε(a)⊆A for some ε>0}
(xn) converges to a point x∈X
ε>0, ∃N: xn∈Bε(x) whenever n>=N
(xn) Cauchy
ε>0, ∃N: d(xm,xn) < ε whenever m,n>=N
d1, d2 on X “Lipschitz equivalent”
⟹
∃h,k positive constants:
∀x,y∈X, hd2(x,y)<=d1(x,y)<=kd2(x,y)
d1,d2 “topologically equivalent”
A⊆X d1-open in X
⟺
A⊆X d2-open in X
isometry
f:(X,dx)->(Y,dy)
f bijective such that
dy(f(x1),f(x2)) = dx(x1,x2), ∀x1,x2∈X
A⊆X dense in X
A_ = X
x point of closure of A⊆(X,d)
ε>0, Bε(x)∩A ≠ ∅
f:X->Y “uniformly continuous” on X
ε>0, ∃δ>0:
dy(f(x),f(x’)) < ε ⟹ dx(x,x’)<δ
∀x,x’∈X
X compact, f continuous
Lebesque number
- 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.
ε-net for X
N⊆X : {B↓(ε) (x): x∈N} covers X, some ε>0
X “complete”
if every Cauchy sequence in X converges (to a point of X)
f:X->X “contraction”
d(f(x),f(y))<=Kd(x,y), ∀x,y∈X.
for some constant K<1
Banach’s Fixed Point Theorem
f:X->X contraction, metric space X complete ⟹ f has a unique fixed point p in X
x∈X is in boundary of A⊆X
∀ε>0, A∩Bε(x) and (X\A)∩Bε(x) non-empty
Cauchy (xn) converges to a point x∈X
if (xn) has subsequence converging to x
X “sequentially compact”
every sequence in X has at least one convergent subsequence in X
⟺
X compact
X “sequentially compact”
⟹
- any open cover for X has lebesque number
- X has finite ε-net
X “regular” / “normal”
given V⊆X closed and x∈X\V,
∃U,U’⊆X open disjoint:
V⊆U and x∈U’
f : (X,Tx) -> (Y,Ty) continuous
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
f : (X,Tx) -> (Y,Ty) continuous at a point x∈X
if, given any V∈Ty : f(x)∈V,
there is some U∈Tx: x∈U and f(U)⊆V
basis for T
subfamily B⊆T :
every set in T is a union of sets from B.
X “connected”
X admits no partition
partition {A,B} of X
A,B ⊆ X open, non-empty:
X = A∪B, A∩B=∅
A path f “joins” x and y in X
a continuous map f:[0,1]->X such that f(0) = x and f(1)=y
X “path-connected”
if any two points of X can be joined by a path in X
⟹ X connected
X⊆R^n open connected ⟹
X path-connected
homeomorphism
bijective map f :
f and f^(-1) continuous
A⊆X “compact”
if every open cover for A has a finite sub cover
A⊆X “relatively compact”
if A_ compact in X
Heine-Borel theorem
any closed bounded subset of R^n is compact
connected A⊆X
A ⊆B ⊆A_ ⟹
B connected
f has IVP
f(a)<x<f(b)
∃c ∈[a,b] : f(c) = x
f:D->R satisfies Lipschitz condition of order α on D
α,K>0,
|f(x) - f(y)|<= K|x-y|^α, ∀x,y∈D.
k constant
Cauchy’s criterion for uniform convergence
(fn) converges uniformly on D
⟺
(fn) uniformly Cauchy on D
(fn) “uniformly Cauchy” on D⊆R
if given ε>0, ∃ N∈N:
|fm(x) - fn(x)|<ε , ∀m,n>=N and ∀x∈D
(fn) converges uniformly to f on D
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 -> ∞
(fn) converges pointwise on D
(fn(x)) converges to f(x), ∀x∈D
Bolzano-Weierstrass theorem
every bounded sequence of real numbers has at least one convergent subsequence
Sup(X)
the upper bound s is supremum if ∀ε>0, ∃x∈X : s-ε < x
axiom of completeness
any non-empty set of real numbers that is bounded above has a least upper bound
