2012-13 L3 MAF Topologie & analyse hilbertienne Feuille 1

X
(Ai)i∈I (Bj)j∈J X
(∪i∈IAi) ∩ (∪j∈JBj) = ∪(i,j)∈I×J(Ai ∩ Bj) (∩i∈IAi) ∪ (∩j∈JBj) = ∩(i,j)∈I×J(Ai ∪ Bj)
((Aj)j∈Ji )i∈I X
∪i∈I(∪j∈Ji Aj) = ∪j∈∪i∈IJi Aj ∩i∈I(∩j∈Ji Aj) = ∩j∈∩i∈IJi Aj
(Ai)i∈I (Bi)i∈I X
(
Ai) ∩ (
Bi) =
(Ai ∩ Bi)
i∈I
i∈I
i∈I
X, Y f : X → Y
A X f(A) = {f(x) x ∈ A} = {y ∈ B | ∃x ∈ A, y = f(x)}
B Y f −1 (B) = {x ∈ A | f(x) ∈ B}
∀A ∈ P(X), A ⊂ f −1 (f(A))
f injective ⇔ ∀A ∈ P(X), f −1 (f(A)) = A.
∀B ∈ P(Y ), f(f −1 (B)) ⊂ B
f surjective ⇔ ∀B ∈ P(Y ), f(f −1 (B)) = B.
(Ai)i∈I X
f(∪i∈IAi) = ∪i∈If(Ai) f(∩i∈IAi) ⊂ ∩i∈If(Ai) f
(Bi)i∈I Y
f −1 (∪i∈IBi) = ∪i∈If −1 (Bi) f −1 (∩i∈IBi) = ∩i∈If −1 (Bi)
A ∈ P(X) B ∈ P(Y ) f −1 (B c ) (f −1 (B)) c f(A c ) (f(A)) c
A, B A × B
z
z
{0, 1} N Y X
X Y Y N Y
(E, d) δ(A) = sup x,y∈A d(x, y)
A E δ(∅) = 0
δ(A) = 0 A E δ( Ā) = δ(A)
R Ai Ai (Ai)
A1 = [−3, 1[∪]1, 2] ∪ {3, π} A2 = Z A3 = {(−1) p + 1/2p | p ∈ Z} A4 = Q A5 =] − ∞, 3/2] ∩ Q

R R G
(R, +) G = aZ a
G R R G
G + = {g ∈ G , g > 0} m = inf G +
m > 0 m ∈ G + G = mZ
m = 0 m ∈ G + G R
a, b b = 0 a
b /∈ Q Ga,b := aZ + bZ
R R
a /∈ Q {cos(na) , n ∈ Z} [−1, +1]
π
Ga,2π
R 2 R 2
A = (] − ∞, −1] × {0}) ∪ ([−1, 1[×[−1, 1[) ∩ (R ∗ × R) B = Q × (R \ Q)
C = {(x, y) ∈ R 2 x 2 + y 2 ≤ 2, (x − 1) 2 + y 2 > 1} D = {(x, y) ∈ R 2 x 2 − sin y ≤ 4}
E = {(x, y) ∈ [0, 1] 2 cos x > 0} k ∈ R ∗ + Fk = ∪n∈N ∗Hk,n Fk,n
(1/n, 1/n) k/n
A X
A X
(X, d) Y
X f : Y → R k ∀x, y ∈ Y |f(x) − f(y)| ≤
k d(x, y) x ∈ X y ∈ Y fy(x) = f(y) + k d(x, y)
x ∈ X {fy(x)|y ∈ Y } f(z) − k d(z, x)
z ∈ Y g(x)
x ∈ Y g(x) = fx(x)
g : X → R k
f X
Y X
E (Ai)i∈I
E ∪i∈I Ai (∪i∈IAi) E = R 2 I = N ∗ Ai = {(1/i, 1/j) | j ∈
N ∗ }
(∩i∈IAi) ⊂ ∩i∈I Ai
A
A (A) A
A
A A ∩ (A) = ∅
A (A) ⊂ A
A (A) = ∅
A ⊂ (A) ( A) ⊂ (A) R
A B
(A ∪ B) ⊂ (A) ∪ (B)
R
Ā ∩ ¯ B = ∅ ⇒ (A ∪ B) = (A) ∪ (B)