TD : feuille n°1 Rappels de topologie, construction d`espaces

E P
F⊂E P
P E
F
(Fi)i∈IE P
Si∈IFiTi∈IFiP
P
E
X Y X Y
f:X→Y
Xe X=X∪ {∞}
Xe X X
e
X
e
Xe X\ {∞}
X
e
X
e
X
X→e X\ {∞}
f
RnSn

K K =R C KPn
Kn+1 \ {0}K∗(x0, . . . , xn)∈
Kn+1\{0}[x0:. . . :xn]KPn
{±1}R∗SnSn/{±1}
RPnRPn
¯
Dnx∈∂¯ Dn−x
S1C1S2n+1 ⊂Cn+1
S2n+1/S1CPn
RP1S1CP1S2
n≥1
Kn→KPn
(x1, . . . , xn)7→ [1 : x1:. . . :xn]
KnU0KPn
KPnn+ 1 Kn
KPn−1
T[0,1]×[0,1] (x, 0) ∼(x, 1)
(0, y)∼(1, y)x, y ∈[0,1]
T
S1×S1
R2Z2
R3R3
{y= 0}(2,0,0) 1 (Oz)
X A ⊂X X/A X
x∼y x, y ∈A x =y.
n∈N∗¯ Dn/∂ ¯ DnSn
X A X/A
X A X X/A
(Xi, xi)i∈I∨i∈IXi
XiXixi
n∈N∗Sn−1Sn
(x1, . . . , xn)7→ (x1, . . . , xn,0) Sn/Sn−1n
H⊂R2
(1/n, 0) 1/n n ≥1 (S1
i)i∈N
S1∨i∈NS1
iH

C n C
f:¯ Dn→X C n ¯ DnX
f Dn¯ DnDnC
¯
Dn< n
A⊂X A
X
n= 0 ¯ D0=D0X
X n
n X
n X X
k≥0k XkX
k X X
Y⊂X X
n≥1SnSn
k Skk∈ {0, . . . , n}
RPnCPnR
X C n X
f:¯ Dn→X C C X f(¯ Dn)
X k ≥0k XkX
X
X k ≥1Xk/Xk−1
k
n≥1p≥0x0, . . . , xp∈Rn
x1−x0, . . . , xp−x0pRn
p+ 1 x0, . . . , xpRn
p
Rn
KRn
K K
σ, σ0∈K
σ σ0
|K|K
RnRn
KRn
K|K|