Definition 5.4.1. A topological space is quasicompact if every open cover has a finite subcover. A topological space is noetherian if any subset is quasicompact in its induced topology. Remark 5.4.2. The following conditions
are equivalent for a topological space:
In a noetherian topological space X, every nonempty closed subset Y can be expressed as a finite union Y = È_{i} Y_{i} of irreducible closed subsets Y_{i}. The Y_{i} are uniquely determined if Y_{i} Ë Y_{j} for i ¹ j; they are called the irreducible components of Y. A topological space is locally noetherian if any point has a neighborhood which is a noetherian space in its induced topology. Let C be a metric site with finite limits. Definition 5.4.3. A morphism f: X ® Y is called quasicompact if for any open quasicompact subset U of Y, f^{1}(U) is quasicompact (in this case we also say that X is quasicompact over Y). Definition 5.4.4. A morphism f: X ® Y is called quasiseparated (or X is a Yobject quasiseparated over Y) if the diagonal morphism D_{f}: X ® X ×_{Y} X is universally quasicompact. We say that X is quasiseparated if X is quasiseparated over the final object Z. Remark 5.4.5. (a) The composite of two
quasicompact morphisms is quasicompact.
Proposition 5.4.6. If X is noetherian, then any morphism f: X ® Y is quasicompact. Proof. Any subset of a noetherian space is quasicompact, therefore f^{1}(U) is quasicompact for any quasicompact subset U of Y. Proposition 5.4.7. Suppose f: X
® Y is a bicontinuous morphism. Then
f is quasicompact if any of the following conditions is satisfied:
Proof. Since f is bicontinuous, we can identify X
as a subspace of Y.
Proposition 5.4.8. Any separated morphism is quasiseparated. In particular, any separated object is quasiseparated. Proof. If f is separated, D_{f} is universally closed, hence universally quasicompact by (5.4.7.a). Proposition 5.4.9. Suppose f: X
® Y is a morphism. Suppose the underlying
space of Y has a basis B consisting of quasicompact open subsets.
Then
Proof. (a) Suppose the condition is satisfied. Any quasicompact
open subset of Y is a finite union of the open sets in B.
Therefore for any quasicompact open subset V of Y, f^{1}(V)
is a finite union of quasicompact open subsets, hence a quasicompact
set. The other direction is obvious.
Proposition 5.4.10. Suppose f: X ® Y, g: Y ® Z are two morphisms. If gf is quasicompact and f is surjective, then g is quasicompact. Proof. If V is an open quasicompact subset of Z, f^{1}(g^{1}(V)) is quasicompact, hence f(f^{1}(g^{1}(V))) Í Y is quasicompact. But g^{1}(V) = f(f^{1}(g^{1}(V))) as f is surjective. Proposition 5.4.11. (a) The composition
of two quasiseparated morphisms is separated.
Proof. The proofs are similar to those of (5.3.4). Corollary 5.4.12. (a) Suppose X
is quasiseparated. Then any morphism f: X ®
Y is quasiseparated.
Proof. Applying (5.4.11.a and d) to X, Y and the final object Z. Proposition 5.4.13. Suppose f: X ® Y is a morphism, {U_{i}} is an effective open cover of Y such that each subobject U_{i} is quasiseparated. Then f is quasiseparated if and only if any subobject X_{i} = f^{1}(U_{i}) is quasiseparated. Proof. For each i denote by f_{i} the restriction
X_{i} ® U_{i}.
Then the inverse image of U_{i} in X ×_{Y}
X under the map X ×_{Y} X
® Y is X_{i} ×
_{Ui} X_{i}, and the restriction X_{i}
® X_{i} ×_{Ui}
X_{i} of D_{f}
is D_{fi}. Then by the definition
of quasiseparated morphism and the local nature of quasicompact morphism,
f is quasiseparated if and only if each f_{i} is
quasiseparated. But by assumption each U_{i} is
quasiseparated, the proof is finished by (5.4.12.b).
