Fiche résumée du cours de topologie 1 Topologies

O ⊂ P(E)
∅ ∈ O E∈ O
∀U, V ∈ O U∩V∈ O
(Ui)i∈IO ∪
i∈IUi∈ O
(E, O)E
OE(E, O)O
(E, O)F E −F∈ O
(E, O) (E0,O0)
E×E→R+
∀x∈E, d(x, x)=0
(E, d)
(E, d) (E0, d0)k≥0
f:E→E0
∀x, y ∈E d(f(x), f(y)) ≤k.d(x, y)
f:E→E0
∀x, y ∈E d(f(x), f(y)) = d(x, y)
(E, d)O ⊂ P(E)
U∈ O ⇔ ∀a∈U, ∃r > 0|B(a, r)⊂U
(E, O)
(E, d)Id →(E, d0)
∃c > 0∀(x, y)∈E2
c−1d(x, y)≤d0(x, y)≤c.d(x, y)

d(x, A) = inf
y∈Ad(x, y)
Vr(A) = {x∈E, d(x, A)< r}
sup
y∈A
x∈A
d(x, y)
k.kd(x, y) =kx−yk
(EJ, dJ)J= 1..N E =
E1×E2×.. ×ENp∈[1; +∞[x= (x1..xn), y = (y1..yn)∈E
dp(x, y)=(
N
X
J=1
dJ(xJ, yJ)p)1/p
d∞(x, y) = max
J=1..N dJ(xJ, yJ)
(di)
di(x, y)=0∀i⇒x=y
k0k= 0
kxk= 0 ⇒x= 0
x∈Ekxki= 0 ∀i
(di)i∈I
O ⊂ P(E)
U∈ O ⇔ ∀a∈U, ∃r > 0,∃J⊂I, , | ∩
i∈JBdi(a, r)⊂U
Σ = {(Ai)i∈I}
J⊂I AJ=∩
i∈JAiO ⊂ P(E)
AJJ⊂I
O
Σ Σ O
(E, O)
OBO
B
BO ⇔ B⊂ O ∀U∈ O,∀x∈
U, ∃B∈B|x∈B⊂U
∀P1, P2∀x∈P1∩P2
∃P0⊂P1∩P2, x ∈P0
(E, O)B⊂ P(E)
∪
B∈BB=E∀U, V ∈B∀x∈U∩V∃ω∈B, ω ⊂U∩V x ∈ω
B

(E, O)a∈E V
a∈U⊂V V
A⊂E A ⊂U⊂V
F⊂ P(X)
F
U∈FU⊂V V ∈F
∅ ∈ F
x∈X X ⊂ P(X)
B⊂ P(X)
B
∃B∈BB⊂U
(E, O)A⊂E
˚
A
A
fr(A) = ∂A =A−˚ A=A∩(E−A)
(E, O)A⊂E
˚
A˚
˚
A=˚ A
A A =A
˚
A∩B=˚ A∩˚ B A ∪B=A∪B
˚
A∪˚ B⊂˚ A∪BA∩B⊂A∩B
E−˚ A=E−A E −A=˚ E−A
A⊂E A =E
∀(x, y)∈E2∃U, V
x∈U y ∈V
∀x∈E, {x}

f:E→F a ∈E
f a ∀V f(a)∃U a |f(U)⊂V
∀V f(a)f−1(V)
f:E→F
∀V F f−1(V)E
∀V F f−1(V)E
∀V F f−1(V)E
∀V F f−1(V)E
∀a∈E f
∀A⊂E f(A)⊂f(A)
f
f:E→F
f(A)
f:E→F f
f f−1
f
f
f
f(U)f(F)
f:F→R
g:E→R
f
f:E→[0,1] f/F =e 1f/E−U=e 0
f:E→[0,1] f/F =e 1
f/G =e 0

A⊂E A =∅A=E
f:E→X
f:E→ {0,1}
∀x, y ∈E
∃γ[0; 1] →E γ(0) = x γ(1) = y
O1O2O1
O2O2⊂ O1
O1O2
O2O1
O2O1
∀x∈EO2O1
Vj(x)Oj∀x∈E
∀V2∈ V2(x)∃V1∈ V1(x)V1⊂V2
(Yi,Oi)i∈I
fi:X→Yi(fi)i∈I
(fi)i
(fi)i
(di)i∈I
(di(., y))i∈I,y∈E
f:Z→X
(fi)if
fi◦f
(di)i∈I