Exercise 13, § 4

Let A be a ring and X = Spec(A) its spectrum. (a) If a subset F of X is closed, it has the following two properties: (α) for all p ∈ F, V(p) ⊂ F; (β) for all p ∉ F, there exists a closed subset of V(p) which contains F ∩ V(p) and does not contain p. (b) Suppose X is Noetherian. Show that every subset F of X with properties (α) and (β) of (a) is closed in X (consider the irreducible components of F̄ and use Exercise 9 (a)). (c) With the notation of Exercise 12, take Y to be the interval [0, 1] of R. Show that Y is not closed in X = Spec(A) (cf. § 2, Exercise 15) but satisfies conditions (α) and (β) of (a).