2.5 Irreducible Schemes
Let \(|X|\) be a topological space.
\(|X|\) is irreducible if it cannot be written as a union of two closed subsets \( Z_1 \cup \mathbf {Z}_2\).
Let \(X\) be a suitable (noetherian? irred? zariski?) topological space. Then the intersection of an open with an irreducible is irreducible.
Now, let \(X\) be a scheme with \(|X|\) its associated topological space.
\(X\) is irreducible if \(|X|\) is.
Let \(X\) be a scheme, \(U\) an affine open neighborhood and \(Y\) a closed (irreducible?) subscheme. Then \(U \cap Y\) is an affine open neighborhood of \(Y\)
Let \(X = \operatorname {Spec}R\) be an affine scheme. Then the irreducible subsets of the topological space \(|X|\) are in one-to-one correspondence with the prime ideals of \(R\) on the association \(\mathfrak {p} \mapsto V(\mathfrak {p})\).
This statement looks a lot like the nullstellensatz, but it actually just follows straight from the definitions.