Suppose X is a topological space. If Y is a subset of X, we denote by Y ^{} the closure of Y. A generic point for an irreducible closed subset Z is a point z such that Z = {z}^{}. A topological space is sober if every closed irreducible set has a unique generic point. If x, x' are two points of X and x' Î {x}^{}, then we say that x' is a specialization of x, and x is a generalization of x', written x' £ x. The underlying set of X is a partial ordered set (i.e., a poset) with the partial order £. Any continuous map of topological spaces preserves the partial order. A point x is called a maximal (resp. minimal) point if it is a maximal (resp. minimal) element in the poset X. A subset Y of a space X is called stable (under specialization) if y Î Y and x £ y implies x Î Y. Any closed set is a stable set. For any subset Y we let Y_ = {x Î X  x £ y for some y Î Y} be the stable set generated by Y. Clearly we have Y_ Í Y ^{}. We say Y is preclosed if Y_ = Y ^{}. Definition 6.2.1. A continuous map f: Y ® X of topological spaces is called stable (resp. preclosed) if for any closed subset V of Y, f(V) is stable (resp. preclosed). Remark 6.2.2. A subset Y of X is closed if and only if Y is both stable and preclosed. A continuous map f: Y ® X is a closed map if and only if f is both stable and preclosed. Remark 6.2.3. (a) Any finite union of
preclosed subsets of X is preclosed.
Remark 6.2.4. Suppose f: X
® Y is a continuous map of topological
spaces. Consider the following conditions on f:
Definition 6.2.5. An affine
spectral site is an affine site B having the following
properties:
A spectral site is an algebraic site C with an affine basis B which is an affine spectral site. (Note that a spectral site also satisfies (a) and (b)). Example 6.2.6. We show that ASch
and GASch are affine spectral sites.
Example 6.2.7. Sch and GSch are spectral sites. Proposition 6.2.8. Let f: Y
® X be a quasicompact morphism in a
spectral site. Then
Proof. The assertions (b) follows from (a) by (6.2.2). To prove (a) it suffices to prove that the intersection U Ç f(X) for any affine open subset U is a preclosed subset in U (6.2.3.b). Since f is quasicompact, f^{1}(U) is a finite union of affine open subsets V_{i}. Each f(V_{i}) Í U is preclosed since V_{i} ® U is a morphism of affine objects (6.2.5c). Thus the finite union U Ç f(X) = È f(V_{i}) is a preclosed subset of U (6.2.3.a). An object X in a spectral site C is called a locality if X has exactly one closed point T_{X} and any other point is a generalization of T_{X}. A morphism f: X ® Y of localities is a morphism f from X to Y such that f(T_{X}) = T_{Y}. Example 6.2.9. For any point x of an object X, the effective subobject x^{o} of X is a locality. If f: X ® Y is a morphism, then f induces a morphism of localities x^{o} ® f(x)^{o}. Note that a locality must be affine because X is covered by affine open subsets and the only open neighborhood of T_{X} is X. A morphism f: X ® Y
of irreducible objects in a spectral site is called birational
if f sends the generic point K_{X} of X
to the generic point K_{Y} of Y, and the induced
morphism K_{X} ® K_{Y}
of spots is an isomorphism. (Note that in a spectral site a generic point
x of an irreducible object is effective as we have {x} =
{x}^{o}.)
