Definition 3.18 The algebro-geometric closure of a subset S ⊂ An(k) is defined
S = {a ∈ An(k) : f (a) = 0 for each f ∈ I (S)} = V(I (S)).
Asubset S ⊂ An(k) is closed if S = S;U ⊂ An(k) is open if its complementAn(k) \ U
is closed in An(k).
S = {a ∈ An(k) : f (a) = 0 for each f ∈ I (S)} = V(I (S)).
Asubset S ⊂ An(k) is closed if S = S;U ⊂ An(k) is open if its complementAn(k) \ U
is closed in An(k).
The Zariski topology on affine space induces a topology on subsets V ⊂ An(k):
Z ⊂ V is closed if Z = V ∩ Y for some closed Y ⊂ An(k). If V ⊂ An(k) is an affine
variety then closed subsets of V are precisely closed subsets of An(k) contained in
V. This is called the Zariski topology on the affine variety.
Z ⊂ V is closed if Z = V ∩ Y for some closed Y ⊂ An(k). If V ⊂ An(k) is an affine
variety then closed subsets of V are precisely closed subsets of An(k) contained in
V. This is called the Zariski topology on the affine variety.
Definition 3.21 A function of topological spaces f : X → Y is continuous if for
each closed Z ⊂ Y the preimage f −1(Z) = {x ∈ X : f (x) ∈ Z} is closed.
The concept of a ‘Zariski continuous’ function is really too weak to be of much use.
For instance, any bijective function C → C is automatically Zariski continuous!
One useful class of Zariski continuous functions are the morphisms introduced in
Chapter 1:
each closed Z ⊂ Y the preimage f −1(Z) = {x ∈ X : f (x) ∈ Z} is closed.
The concept of a ‘Zariski continuous’ function is really too weak to be of much use.
For instance, any bijective function C → C is automatically Zariski continuous!
One useful class of Zariski continuous functions are the morphisms introduced in
Chapter 1:
Proposition 3.22 Let φ : An(k) → Am(k) be a morphism of affine spaces. Then
φ is Zariski continuous.
φ is Zariski continuous.
