Cours

E
d:E×E→R+
∀(x, y)∈E×E d(x, y) = d(y, x)
∀(x, y)∈E×E d(x, y) = 0 ⇔x=y
∀(x, y, z)∈E3d(x, y) + d(y, z)≥d(x, z)
R2
{0,1}
d(σ, τ) = (1
2)kσ=ασ0τ=ατ0
α k

x0
BO(x0, r) = {x∈E/d(x0, x)< r}
x0E
x0
x0V(x0)
E O E
O
∅
E
X
XOX
∅, E O
Ai∈ O ⇒ Si∈IAi∈ O
A, B ∈ ⇒ ATB∈ O

X={a, b}
X={a, b, c}
V x0X
x0
A
A=]0,3] S]4,5[ S([5,6[ TQ)S{7}
A
x A U X x
A
A Int(A)A
x A U X x
A
A Adh(A)A
F r(A)

AR
Int(A)Adh(A)F r(A)
A
A X
A
A
Rn
(xn)n∈N
∀ε > 0,∃N > 0, n > N ⇒d(xn, l)< ε
l ε
(xn)n∈N
∀ε > 0,∃N > 0, n > N, p > N ⇒d(xn, xp)< ε

R Q