Topologie et calcul différentiel : exercices corrigés

(X, d)x, y, z ∈X|d(x, y)−d(x, z)|6d(y, z)
(X, d)
Rn
n∈N∗d1, d2, d∞:Rn×Rn→R+x= (x1, . . . , xn), y = (y1, . . . , yn)∈Rn
d1(x, y) =
n
P
i=1
|xi−yi|d2(x, y) = sn
P
i=1
(xi−yi)2d∞(x, y) = max{|xi−yi| | i∈J1, nK}
d1d2d∞Rnd2
Rn
n= 2
R
R
a, b ∈Ra6b
[a, b] ]−∞, a] [a, +∞[R
a<b [a, b[R
N Z Q RrQ R
(X, d)A, B X
A⊂B˚ A⊂˚ B A ⊂B
Int(XrA) = XrA X rA=Xr˚ A
(X, d)A X B
XFr(B) = B∩XrB
Fr(A)X
Fr(A) = A∩Xr˚ A=Ar˚ A
Fr(A) = ∅A X
A X A ∩Fr(A) = ∅
A X Fr(A)⊂A
A X Fr(A) Fr[Fr(A)] = Fr(A)

A⊂XFr[Fr[Fr(A)]] = Fr[Fr(A)]
(X, d)A X
A=X
Int(XrA) = ∅
U X U ∩A6=∅
A⊂X X
(X, d)δ(∅)=0 A⊂X
δ(A) = sup{d(x, y)|x, y ∈A}
A+∞A, B ⊂X
δ(A) = 0 A
δ(A)6δ(B)A⊂B
δA=δ(A)
δ(A∪B)6δ(A) + δ(B)A∩B6=∅
δ(A)<+∞a∈A{d(a, x)|x∈A}
(X, d)
∅X X
X∅X
X
a, a0∈X r, r0>0 B(a, r) = B(a0, r0)a=a0r=r0
X X
X X
X X
(X, d)
X
X
X
X
(X, d)
x, y ∈X
d(x, y) = 0x=y
1x6=y
d X
(X, d)

X=Zd(x, y) = |x−y|x, y ∈X(X, d)
Z
d0:Z×Z→R+,(p, q)7→ 0p=q
1p6=qd1:Z×Z→R+,(p, q)7→ |p−q|
d0d1Z
C > 0d1(p, q)6Cd0(p, q)p, q ∈Z
(X, d)a∈X r > 0 B(a, r)⊂Bf(a, r) B(a, r)⊂
Int[Bf(a, r)]
(X, d)n∈N∗A1, . . . , AnX
A1∪ · · · ∪ An=A1∪ · · · ∪ An
A1∩ · · · ∩ An⊂A1∩ · · · ∩ An
(X, d)A, B X A ∩B=A∩B=∅
A∪B X A B X
(X, d)U X
U X
A⊂X A ∩U=A∩U
(X, d)A X B A
B A B X
A X B A X
(X, d)d
x, y, z ∈X d(x, z)6max{d(x, y), d(y, z)}
X x, y, z ∈X
d(x, z)6=d(y, z)d(x, y) = max{d(x, z), d(y, z)}
X
X X
X
(X, d)A⊂X
˚
A6=∅
D⊂X X D ∩A6=∅

x, y, z ∈X d X d(x, y)6d(x, z) + d(z, y)
d(x, y)−d(x, z)6d(z, y) = d(y, z)d(x, z)6d(x, y) + d(y, z)−d(y, z)6
d(x, y)−d(x, z)
u∈Rni∈J1, nKuii u
d1Rn
x, y ∈Rnx=y d1(x, y) = 0 d(x, y)=0
|xi−yi|= 0 iinJ1, nKx=y
x, y ∈Rn|xi−yi|=|yi−xi|i∈J1, nKd1(x, y) = d1(y, x)
x, y, z ∈Rn|xi−zi|6|xi−yi|+|yi−zi|
i∈J1, nKd1(x, z)6d1(x, y) + d1(y, z)
d2Rn(· | ·)
Rnu, v ∈Rn
(u|v) =
n
P
i=1
uivi
u∈Rn(u|u)>0kuk2=p(u|u)u, v ∈Rn
|(u|v)|6kuk2· kvk2
ku+vk26kuk2+kvk2
d2(x, y) = kx−yk2x, y ∈Rn
(· | ·)d2(x, z)6d2(x, y) +
d2(y, z)x, y, z ∈Rnu=x−y v =y−z
Rn
u, v ∈Rnv v
λ∈R
06(u+λv |u+λv)=(u|u)+2λ(u|v) + λ2(v|v)
(u|u)+2X(u|v) + X2(v|v)R[X]
4(u|v)2−4(u|u)(v|v)60
u, v ∈Rn
ku+vk2
2= (u+v|u+v)=(u|u)+2(u|v)+(v|v)6kuk2 2+2kuk2·kvk2+kvk2 2= (kuk2+kuk2)2
d∞Rn
x, y ∈Rnx=y d∞(x, y) = 0 x6=y
j∈J1, nKxj6=yjd∞(x, y)>|xj−yj|>0
x, y ∈Rn|xi−yi|=|yi−xi|i∈J1, nKd∞(x, y) = d∞(y, x)
x, y, z ∈Rnj∈J1, nK|xj−zj|= max{|xi−zi| | i∈J1, nK}
d∞
d∞(x, z) = |xj−zj|6|xj−yj|+|yj−zj|6d∞(x, y) + d∞(y, z)

B1B2B∞R2(0,0) 1 d1d2d∞
(x1, x2)∈B1d1|x1|+|x2|61
x1>0x2>0x261−x1
x1>0x260x2>x1−1
x160x2>0x26x1+ 1
x160x260x2>−1−x1
B1x261−x1x2>x1−1
x26x1+ 1 x2>−1−x1
(x1, x2)∈B2px2
1+x2 261x2
1+x2 261B2
(0,0) 1
(x1, x2)∈B∞d∞max{|xi| | i= 1,2}61
−16xi61i= 1,2B∞
x161x1>−1x261x2>−1
a, b ∈Ra < b ]a, b[
B((a+b)/2,(b−a)/2) a∈R
]a, +∞[R]a, a +n[n∈N∗
]− ∞, a[a∈R[a, +∞[ =
Rr]−∞, a[ ]−∞, a] = Rr]a, +∞[ ]−∞, a[ ]a, +∞[
R[a, +∞[
]−∞, a]R
[a, b] = ]−∞, b]∩[a, +∞[R
a < b a ∈[a, b[ε > 0
a−ε
2∈]a−ε, a +ε[ = B(a, ε)a−ε 2/∈[a, b[
a[a, b[
[a, b[R
Rr[a, b[=]−∞, a[∪[b, +∞[b a
a<b ε>0α= min(ε, b−a)>0α/2/∈Rr[a, b[
ε b Rr[a, b[
Rr[a, b[R[a, b[
R
Q RrQ R a, b ∈Ra<b
]a, b[Q RrQ
N Z R R 0
N=Rr S
n∈N
]n, n + 1[∪]−∞,0[Z=RrS
k∈Z
]k, k + 1[
N Z R
Q R q∈Qε > 0 ]q−ε, q+ε[ = B(q, ε)
RrQx∈RrQ
ε > 0 ]x−ε, x +ε[ = B(x, ε)
Q=Rr(RrQ)RrQ R
˚
A˚ A⊂A A ⊂B˚ A⊂B˚ A X
B˚ A⊂˚ B˚ B
B B ⊂B A ⊂B A ⊂B B X
A A ⊂B A