In this section we assume C is a metric site with finite limits. Let Z be a final object of C. 

Definition 5.3.1. A morphism f: X ® S is called separated if the diagonal morphism D = (1_X, 1_X)_S: X ® X ×_S X is a universally closed morphism. If f is separated we also say that X is separated over S. An object X is called separated if it is separated over the final object Z. 

Proposition 5.3.2. Suppose f: X ® S is a morphism. The following conditions are equivalent: 
(a) f is separated. 
(b) The kernel p: ker (i,j) ® Y of any two S-morphisms i, j: Y ® X is universally closed. 
(c) Any Y-section of Y ×_S X over Y for any S-object Y ® S is universally closed.  
(c') The graph G_f = (1, f): Y ® Y ×_S X of any S-morphism f: Y ® X is universally closed. 

Proof. By (5.2.8). 

Corollary 5.3.3. Suppose C is a site such that any special morphism is bicontinuous. Suppose f: X ® S is a morphism. Then the following conditions are equivalent: 
(a) f is separated.  
(a') The image |D(X)| of X in X ×_S X is a closed subset of X ×_S Y. 
(b) Suppose i, j: Y ® X are two S-morphisms with the kernel p: ker (i,j) d Y. Then |p(ker(i,j))| is a closed subset of Y. 
(c) Suppose g: Y ® S is a base extension and h is a Y-section of X ×_S Y over Y. Then |h(Y)| is a closed subset of X ×_S Y.  
(c') Suppose f: Y ® X is an S-morphism with the graph morphism G_f = (1, f): Y ® Y ×_S X. Then |G_f(Y)| is a closed subset of Y ×_S X. 

Proof. By (5.2.6). 

Proposition 5.3.4. (a) Any monomorphism is separated. 
(b) Any effective morphism is separated. 
(c) The composition of two separated morphisms is separated. 
(d) Suppose f: X ® Y is a separated S-morphism and j: S' ® S is a morphism. Then the S'-morphism f_S' is separated. 
(e) If f: X' ® X and g: Y' ® Y are separated S-morphisms, then the product morphism f × g: X' ×_S Y' ® X ×_S Y is also separated. 
(f) If the composite gf of two morphisms is separated, f is separated. 

Proof. (a) Suppose f: X ® S is a monomorphism. Then X = X ×_S X (4.2.6.h). Thus |D(X)| = |X ×_S X| is a closed subset of |X ×_S X| and f is separated. 
(b) Any effective morphism is a monomorphism. 
(c) Suppose f: X ® Y , g: Y ® Z are two morphisms. It is easy to see that D_X/Z = ((p, q)_Y)D_X/Y 

Corollary 5.3.5. Let f: X ® S be a separated morphism. Suppose U Í |X| and V Í |S| are effective subsets and f(U) Í V. Then U is separated over V. 

Proof. The composite g: U ® S of the effective morphism U ® X and the separated morphism f: X ® S is separated (5.3.4.c). Let h: U ® V be the induced morphism and i: V ® S the inclusion. Then g = ih is separated implies that h is separated (5.3.4.f). 

Corollary 5.3.6. (a) An object X is separated if and only for any object S, any morphism f: X ® S is separated.  
(b) If f: X ® S is a separated morphism and S is separated, then X is separated. 

Proof. (a) The condition is sufficient because we may take f to be the canonical morphism from X to the final object Z. For the necessity let i: X ® Z and j: S ® Z be the natural morphisms to the final object Z. Then we have i = jf. Thus f is separated if i is so by (5.3.4.f). 
(b) If f: X ® S and j: S ® Z are separated, then i = jf is separated by (5.3.4.c). 

Corollary 5.3.7. Suppose X is a separated. Then any effective subobject of X is a separated object. 

Definition 5.3.8. Suppose f: X ® S is a morphism. A diagonal cover of X over S is an effective open cover {U_i} of |X| such that {|U_i ×_S U_i|} is an open cover of |X ×_S X|. 

Remark 5.3.9. An effective open cover {U_i} is a diagonal cover if any of the following conditions are satisfied. 
(a) U_i = f^-1(V_i) for an effective open cover {V_i} of |S|. 
(b) Each pair x, y Î |X| lie in some U_i. 

Proposition 5.3.10. Suppose {U_i} is a diagonal cover of X such that each U_i is separated over S. Then f is separated. 

Proof. Suppose each U_i is separated over S. Then U_i = D_X^-1(U_i ×_S U_i) ® U_i ×_S U_i is a universally closed morphism. Since {|U_i ×_S U_i|} is an open cover of |X ×_S X|, D_f is a universally closed morphism by (5.2.7.e). Thus f is separated. 

Proposition 5.3.11. Suppose C is a site such that the topology t: C ® Top preserves finite limits. An object X is separated if and only if the underlying space of X is Hausdorff.  

Proof. An object X in such a site C is separated if and only if the space of X is separated in the site Top. Thus it suffices to prove the assertion for Top. First we assume X is a Hausdorff topological space. For any (x, y) Î X × Y and (x, y) Ï D(X), we have x ¹ y, therefore we can find two neighborhoods U and V of x and y respectively such that U Ç V = Æ. Then W = p^-1(U) Ç q^-1(V) = U × V is a neighborhood of (x, y) and W Ç D(X) = Æ. This means that D(X) is closed, hence X is separated since D is universally bicontinuous. Conversely, suppose X is separated. Then D(X) is a closed subset of X × X. If x, y are two points of X and x ¹ y, then (x, y) - D(X). Thus we can find a neighborhood W of (x, y) such that W Ç D(X) = Û. By the definition of product topology on X × Y we can find neighborhoods U and V of x and y respectively such that U × V is a neighborhood of (x, y) contained in W. Then we must have U Ç V = Æ. This shows that X is Hausdorff. 

Example 5.3.12. Suppose X is a Hausdorff space. Suppose f: X ® S is a continuous map and g: S ® X is an S-section of X. Then g(S) must be a closed subset (5.3.3.c). This fact can also be checked directly. Regarding S as a subset of X, if x Î X - S then g(x) ¹ x. Since X is Hausdorff, we can find two neighborhoods V and V' of x and g(x) respectively such that V Ç V' = Æ. Let W = V Ç g^-1(V'). Then W is a neighborhood of x and W Ç S = Æ. This means that S is closed.