Corrigé de la feuille n°1 Rappels de topologie, construction d

E
E∅E
N∅0
{0}N N
N=[
n≥0
{0, . . . , n}
N
E∅
E
R R\{0}.
S1
R2x7→ sin( 1
x) ]0,+∞[
{0}×[−1,1]
R2
Q R
Q
x∈QK x α, β ∈R]α, β[∩Q⊂K
K
X
X=(R\Q)× {−1,1}∪Q× {0}
U⊂X U (V× {−1,0,1})∩X V
R R

X
r∈R\Q(r, −1) (r, 1)
XR
A1= (R× {0,1})∩X
A−1= (R× {0,−1})∩X.
A1A−1Q
R
{(0,0)}∪{(x, y)∈R2, x > 0}
(0,0)
R2
A= ({0} × [0,1]) ∪[
n≥1
An
An(0,1
n) ( 1 n,0)
B= ([0,1] × {0})∪[
n≥1
Bn
Bn(0,1
n) (−1 n,0). A B
A∩B={(0,0)}∪{(0,1
n), n ≥1}
SR2
[a, b[×[c, d[Q2
∆y=−x∆
(x, −x) ∆ ∆ ∆
[x, x + 1[×[−x, −x+ 1[.
∆ ∆
∆+={(x, y), y > x}
∆
∆−={(x, y), x > y}
X
X
F X (Ui)i∈IF
i ViX Vi∩F=Ui
{Vi, i ∈I} ∪ {Fc}X
J⊂I F c∪Sj∈IVj=X
F

(Fi)i∈I
E
Fi
f
F X X F
f(F)Y Y f(F)
∅
∅X
U1, . . . , Une X∞
U1X
U1∩. . . ∩UnU1
(Ui)i∈Ie X
∞Xe X
Si∈IUic=Ti∈IUc
iX
Si∈IUie
X
e
X x y e X x y
X X U V X e X
x∈U y ∈V y =∞X
K x X
Ke X x y
Ue X U∞∈ U ∞ K∞
Ufin ⊆ U K∞e X
ι:X→e Xr{∞} e Xr{∞}
X ι
Te XTe X
e
X
id : X →e Xr{∞}
XTV∈ T V
X U ∪ {∞} U X
∞ ∈ V∈ T e XrV⊆e Xr{∞} e X
T ⊆ TAlex
(e X, TAlex)→(e X, T)
Fe X
FT
(e X, T) ( e X, TAlex)→(e X, T)
T=TAlex
Sn
∞SnSnr{∞} ' Rn∞
(1,0,...,0)

X R ⊂X×X Y =X/R
R
π:X→X/R =Y
X R Y
π U ⊂Y π−1(U)
X
Rn+1 \ {0} → RPnSn
x y SnRPnx=±y
Sn/{±1} → RPn.
Rn+1 \ {0} → Sn
(x0, . . . , xn)7→ x0
[[x|| ,..., x ||x|| 
|| · || p:Sn→Sn/{±1}
RPn+1 →Sn/{±1}
RPn
Sn/{±1}CPnS2n+1
Cn+1 R2n+2 CPnS2n+1/S1
S2n+1
x, y ∈S2n+1 S2n+1/S1S1
S1→Rλ7→ ||x+λy|| >0
x y  < 1
2minλ∈S1||x+λy|| x
y
Vx:= (∪λ∈S1B(λx, )) ∩S2n+1
Vy
R2\ {x= 0} → R
(x, y)7→ y
x
RP1\ {[0 : 1]}=R2\ {x= 0}/R∗→R.
R→R2\ {x= 0} → RP1\ {[0 : 1]}
z7→ (1, z)7→ [1 : z]
[0 : 1] RP1R RP1
RP1RS1
CP1C=R2
S2
S1/{±1} ' RP1
S1→S1z7→ z2
S1S1/{±1} ' S1
S3/S1'S2

U0={[x0:. . . :xn]∈KPn, x06= 0}
V0={(x0, . . . , xn)∈Kn+1 \{0}, x06= 0} ∈ Kn+1 \{0}q:Kn+1 \{0} →
KPn. U0
[1 : x1:. . . :xn]
φ0q
Kn→Kn+1 \ {0}
(x1, . . . , xn)7→ (1, x1, . . . , xn)
V0→Kn
(x0, . . . , xn)→x1
x0,...,xn x0,
U0→Knφ0
φ0
Ui={[x0:. . . :xn]∈KPn, xi6= 0}Ui
Kn{U0, . . . , Un}KPn
U0F0={[x0:x1:. . . :xn]∈KPn, x0= 0}
G0={(x0, . . . , xn)∈Kn+1 \ {0}, x0= 0}
q F0=G0/K∗G0Kn\0
(0, x1, . . . , xn) (x1, . . . , xn)
S1R→S1
x7→ e2iπx R/Z'S1R/Z R
Z
S1×S1'R/Z×R/Z'R2/Z2.
Trev S1×S1
S1×S1→Trev
(θ, φ)7→ ((2 + cos φ) cos θ, (2 + cos φ) sin θ, sin φ)
θ(2cosθ, 2 sin θ, 0)
θ(Oz)φ(0,0,sin φ)
2 + cos φ z = sin φ
T T
[0,1]2[0,1]2→S1×S1
(x, y)7→ (e2iπx, e2iπy)T→S1×S1
x= (x1, . . . , xn)∈Dn||x|| x
f:Dn\{0} −→ Sn⊂Rn+1
x7→ x
||x|| sin(π||x||),cos(π||x||).
||x|| → 0 (0,...,0,1) f
Dn→Snx∈∂Dn=Sn−1
(0,...,0,−1) ∈Sncos(π||x||) = −1||x|| =
1f DnSn\{(0,...,0,−1)}.