5.4 Quasi-Compact Morphisms 

Definition 5.4.1. A topological space is quasi-compact if every open cover has a finite subcover. A topological space is noetherian if any subset is quasi-compact in its induced topology. 

Remark 5.4.2. The following conditions are equivalent for a topological space: 
(a) X is noetherian; 
(b) X satisfies the ascending chain condition for open subsets; 
(b') every nonempty family of open subsets has a maximal element; 
(c) X satisfies the descending chain condition for closed subsets; 
(c') every nonempty family of closed subsets has a minimal element; 
(d) every open subset is quasi-compact. 

In a noetherian topological space X, every nonempty closed subset Y can be expressed as a finite union Y = Èi Yi of irreducible closed subsets Yi. The Yi are uniquely determined if Yi Ë Yj 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 quasi-compact if for any open quasi-compact subset U of |Y|, f-1(U) is quasi-compact (in this case we also say that X is quasi-compact over Y). 

Definition 5.4.4. A morphism f: X ® Y is called quasi-separated (or X is a Y-object quasi-separated over Y) if the diagonal morphism Df: X ® X ×Y X is universally quasi-compact. We say that X is quasi-separated if X is quasi-separated over the final object Z

Remark 5.4.5. (a) The composite of two quasi-compact morphisms is quasi-compact. 
(b) If f: X ® Y is quasi-compact, then for any open subset V of |Y|, the restriction f-1(V) ® V of |f| is quasi-compact. 

Proposition 5.4.6. If X is noetherian, then any morphism f: X ® Y is quasi-compact. 

Proof. Any subset of a noetherian space is quasi-compact, therefore f-1(U) is quasi-compact for any quasi-compact subset U of Y

Proposition 5.4.7. Suppose f: X ® Y is a bicontinuous morphism. Then f is quasi-compact if any of the following conditions is satisfied: 
(a) f is a closed morphism. 
(b) Y is a locally noetherian space

Proof. Since f is bicontinuous, we can identify |X| as a subspace of |Y|. 
(a) If U is any quasi-compact open subset of |X|, then f-1(U) = |X| Ç U is a closed subset of U, which is quasi-compact in its induced topology. (b) If Y is locally noetherian, then any open quasi-compact subset V of |Y| is noetherian, hence f-1(V) = |X| Ç V Í V is quasi-compact. 

Proposition 5.4.8. Any separated morphism is quasi-separated. In particular, any separated object is quasi-separated. 

Proof. If f is separated, Df is universally closed, hence universally quasi-compact 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 quasi-compact open subsets. Then  
(a) f is quasi-compact if and only if for any open set V Î B, f-1(V) is quasi-compact,  
(b) If {Vi} is an open cover of |Y| such that any restriction f-1(Vi) ® Vi of |f| is quasi-compact, then f is quasi-compact.  
(c) Suppose f is quasi-compact, and U Í |X|, V Í |Y| are closed subsets such that f(U) Í V. Then the continuous map f' = f|U: U ® V is quasi-compact. 

Proof. (a) Suppose the condition is satisfied. Any quasi-compact open subset of |Y| is a finite union of the open sets in B. Therefore for any quasi-compact open subset V of |Y|, f-1(V) is a finite union of quasi-compact open subsets, hence a quasi-compact set. The other direction is obvious. 
(b) Suppose {Vi} is an open cover of |Y| such that any restriction f-1(Vi) ® Vi is quasi-compact. The open subsets U Î B such that U Í Vi for some i forms a basis for the underlying space of Y, and f-1(U) is quasi-compact if the morphism f-1(Vi) ® Vi is quasi-compact. Thus we may apply (a). 
(c) Suppose f is quasi-compact. For any open set W of B, W Ç V is a quasi-compact open subset of V as V is closed and W is quasi-compact. Since f-1(W) is quasi-compact, f-1(W) Ç U = f-1(W Ç V) Ç U is quasi-compact as U is closed. Such open sets W Ç V for all V Î B form a basis for V, hence we may apply (a). 

Proposition 5.4.10. Suppose f: X ® Y, g: Y ® Z are two morphisms. If gf is quasi-compact and f is surjective, then g is quasi-compact. 

Proof. If V is an open quasi-compact subset of Z, f-1(g-1(V)) is quasi-compact, hence f(f-1(g-1(V))) Í |Y| is quasi-compact. But g-1(V) = f(f-1(g-1(V))) as f is surjective. 

Proposition 5.4.11. (a) The composition of two quasi-separated morphisms is separated. 
(b) Suppose f: X ® Y is a quasi-separated S-morphism and j: S' ® S is a morphism. Then the S'-morphism fS' is quasi-separated.  
(c) If f: X' ® X and g: Y' ® Y are quasi-separated S-morphisms, then the product morphism f ×S g: X' ×S Y' ® X ×S Y is also quasi-separated. 
(d) If the composition g.f of two morphisms f: X ® Y, g: Y ® Z are quasi-separated, then f is quasi-separated.  

Proof. The proofs are similar to those of (5.3.4). 

Corollary 5.4.12. (a) Suppose X is quasi-separated. Then any morphism f: X ® Y is quasi-separated. 
(b) Suppose Y is quasi-separated. Then a morphism f: X ® Y is quasi-separated if and only if X is quasi-separated. 

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, {Ui} is an effective open cover of Y such that each subobject Ui is quasi-separated. Then f is quasi-separated if and only if any subobject Xi = f-1(Ui) is quasi-separated. 

Proof. For each i denote by fi the restriction Xi ® Ui. Then the inverse image of Ui in |X ×Y X| under the map |X ×Y X| ® |Y| is |Xi × Ui Xi|, and the restriction Xi ® Xi ×Ui Xi of Df is Dfi. Then by the definition of quasi-separated morphism and the local nature of quasi-compact morphism, f is quasi-separated if and only if each fi is quasi-separated. But by assumption each Ui is quasi-separated, the proof is finished by (5.4.12.b). 

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