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(X, τ) D ⊆ X X
τ |D = {A ∩ D : A ∈ τ}
(D, τ |D) τ |D X D τ |D
X D B X B |D = {B ∩D : B ∈ B}
(X, τ)
T0 x, y ∈ X x y
y x
T1 x, y ∈ X U V x ∈ U y /∈ U
y ∈ V x /∈ V
T2 x, y ∈ X U V
x ∈ U y ∈ V
R n
R ∅ R {x ∈ R : x > d}
d ∈ R T0 T1
R ∅ R R
T1 T2
V ∞ R n R n \ V
D ⊆ R n x0 ∈ R n ∪ {∞} D
f : D → R m ℓ ∈ R m ∪ ∞ x x0 f ℓ
V ℓ f −1 (V ) := {x ∈ D : f(x) ∈ V } x0 D
R n
lim
x→x0
x∈D
f(x) = ℓ
x0 ∈ R n ℓ ∈ R m ε > 0
δ > 0 x − x0 < δ x ∈ D f(x) − ℓ < ε
lim
lim
x→x0
x∈D
x−x0→0 +
x∈D
f(x) = ℓ,
f(x) − ℓ = 0.
dℓ∞ d ℓ = (ℓ1,
(f1, ..., fm)
..., ℓm) f =
lim f(x) = ℓ lim fj(x) = ℓj ∀j = 1...m.
x→x0
x∈D
x→x0
x∈D
x0 = ∞ ℓ ∈ Rm ε > 0 M > 0 x > M x ∈ D
f(x) − ℓ < ε
lim f(x) = ℓ,
x→∞
x∈D