The constructible topology on spaces of valuation domains

We consider properties and applications of a compact, Hausdorff topology called the"ultrafilter topology"defined on an {\sl arbitrary spectral space} and we observe that this topology coincides with the constructible topology. If $K$ is a field and $A$ a subring of $K$, we show that the space Zar$(K|A)$ of all valuation domains, having $K$ as quotient field and containing $A$, (endowed with the Zariski topology) is a spectral space by giving in this general setting the explicit construction of a ring whose Zariski spectrum is homeomorphic to Zar$(K|A)$. We extend results regarding spectral topologies on the spaces of all valuation domains and apply the theory developed to study representations of integrally closed domains as intersections of valuation overrings. As a very particular case, we prove that two collections of valuation domains of $K$ with the same ultrafilter closure represent, as an intersection, the same integrally closed domain.


Introduction
The motivations for studying spaces of valuation domains come from various directions and, historically, mainly from Zariski's work for building up algebraic geometry by algebraic means (see [30] and [32]), from rigid algebraic geometry started by J. Tate (see [29], [14], and [20]) and from real algebraic geometry (see [27] and [20]); for a deeper insight on this topics see the paper by Huber-Knebusch [21].
Let K be a field and let A be a subring of K. The goal of this paper is to extend results in the literature concerning topologies on the collection of valuation domains which have K as quotient field, and which have A as a subring and to provide some applications of these results to the representations of integrally closed domains as intersections of valuation overrings. We denote this collection by Zar(K|A). In case A is the prime subring of K, then Zar(K|A) includes all valuation domains with K as quotient field and we denote it by simply Zar(K). A first topological approach to the space Zar(K) is due to Zariski that proved the quasi-compactness of this space, endowed with what is now called the Zariski topology (see [31] and [32]). Later, it was proven, and rediscovered by several authors with a variety of different techniques, that if K is the quotient field of A then Zar(K|A) endowed with Zariski's topology is a spectral space in the sense of Hochster [17] (see [3], [4], [21] and the appendix of [22]). In Section 2, we start by recalling the definition and the basic properties of the constructible topology on an arbitrary topological space, using the notation introduced in [28, Section 2] (for further information cf. [2, §4], [16, (I.7.2.11) and (I.7.2.12)], [17]). Then, we provide a description of the closure in the contructible topology of any subset of a spectral space by using ultrafilters and "ultrafilter limit points" (definition given later). As an application, we obtain a new proof that the ultrafilter topology on the prime spectrum of commutative ring R, introduced in [13], is identical to the classical constructible topology on this space. Section 3 is devoted to the study of the space Zar(K|A) for any subring A of K, endowed with the Zariski topology or the constructible topology. The versatility of the ultrafilter approach to the constructible topology is demonstrated in this section, and in the following Section 4, where we make use of Kronecker function rings. The key result in Section 3 is a proof that the space Zar(K|A) is spectral with respect to the constructible (and to the Zariski) topology by giving, in this general setting, the explicit construction of a ring whose prime spectrum is canonically homeomorphic to Zar(K|A). This is broader than the results of Dobbs, Fedder, and Fontana (cf. [3] and [4]), who proved their results in the case where K is the quotient field of A (and only considering the case of the Zariski topology).
Especially noteworthy in Section 4 are the applications of the topological properties of Zar(K|A), endowed with the constructible topology (or, with the inverse topology, in the sense of Hochster [17]), to the representations of integrally closed domains as intersections of valuation overrings. For example, Proposition 4.1 indicates that two collections of valuation domains with the same constructible closure will represent the same domain. Similarly, Corollary 4.15 indicates how the constructible topological structure of a collection of valuation domains determines the associated finite-type e.a.b. semistar operation. We also apply these results to the class of vacant domains (those domains which have a unique Kronecker function ring). In particular, Corollaries 4.10 and 4.11 use the constructible topology to characterize vacant domains. We then relate closure in the inverse topology to closure in the constructible topology and restate our results concerning e.a.b. semistar operations in terms of the inverse topology. For some distinguished classes of domains, other important contributions on this circle of ideas were given for instance in [23], [24], [25], and [26].

Preliminaries, Spectral Spaces and Ultrafilter Limit Points
If X is a set, we denote by B(X) the collection of all subsets of X, and by B fin (X) the collection of all finite subset of X. Moreover, if G is a nonempty subset of B(X), then we will simply denote by G (resp. G ) the set obtained by intersection (resp. union) of all the subsets of X belonging to G , i.e., G : Recall that a nonempty collection F of subsets of X is said to be a filter on X if the following conditions are satisfied: Let F (X) be the set of all filters on X, partially ordered by inclusion. We say that a filter F on X is an ultrafilter on X if it is a maximal element in F (X). In the following, we will denote the collection of all ultrafilters on a set X by β(X).
For each x ∈ X, it is immediately seen that β x X := β x := {Z ∈ B(X) | x ∈ Z} is an ultrafilter on X, called the trivial (or fixed or principal ) ultrafilter of X centered on x.
Recall that a spectral space is a topological space homeomorphic to the prime spectrum of a ring, equipped with the Zariski topology. The spectral spaces were characterized by Hochster in 1969 as quasi-compact Kolmogoroff topological spaces, with a quasi-compact open basis stable under finite intersections and such that every nonempty irreducible closed subspace has a generic point [17,Proposition 4].
Let X be a topological space. With the notation used in [28, Section 2] we set: i.e., K(X ) is the smallest subset of B(X ) containingK(X ) and closed with respect to finite ∪, ∩, and complementation. As in [28], we call the constructible topology on X the topology on X having K(X ) as a basis (for the open sets). We denote by X cons the set X equipped with the constructible topology and we call constructible sets of X the elements of K(X ) (for Noetherian topological spaces, this notion coincides with that given in [2, §4]) and proconstructible sets the closed sets of X cons . Now consider Y a subset of X . In the following, we denote by Cl(Y) (respectively, Cl cons (Y)) the closure of Y, with respect to the given topology (respectively, the constructible topology) on X .
Assume that X is a spectral space. In this case, the setK (:=K(X )) is a basis of the topology on X and it is closed under finite intersections. The constructible topology on X is the coarsest topology for whichK is a collection of clopen sets and X cons is a compact, Hausdorff topological space.
We can consider on X the usual partial order, defined by If Y is a subset of X , set Y sp := {x ∈ X | y x, for some y ∈ Y} , Y gen := {x ∈ X | x y, for some y ∈ Y} .
Then Y sp (respectively, Y gen ) is the closure under specializations (respectively, the closure under generizations or the generic closure) of Y.
Following [17], we can also endow the spectral space X with the so called inverse topology (or dual topology), that is the topology whose basis of closed sets is the setK(X ) of all open and quasi-compact subspaces of X (with respect to the given spectral topology). We denote by X inv the set X , endowed with the inverse topology. By [17,Proposition 8], X inv is a spectral space and its constructible topology is clearly equal to the the constructible topology associated to the given spectral topology on X . The following fact provides a motivation for the name given to this topology.

2.2.
Remark. Let X be a spectral space and Y be a subset of X . Then, by [ (Y)) sp and Cl inv (Y) = (Cl cons (Y)) gen .

2.3.
Proposition. Let X be a spectral space, Y be a subset of X and U be an ultrafilter on Y. Set Then, the following statements hold.
(1) The set K Y,U is a singleton, the set K Y,U := K Y,U is an irreducible closed subset of X . The generic point of K Y,U is the unique point x U := x Y,U ∈ K Y,U . We will call the point x U ∈ X the ultrafilter limit point of Y, with respect to U .
Proof. (1) By construction K Y,U is a collection of closed subsets of X cons with the finite intersection property. Thus, K Y,U is nonempty. Since X is, in particular, a topological space satisfying axiom T 0 , the conclusion will follow if we Conversely, let z ∈ K Y,U and let U be an open neighborhood of z, with respect to the spectral topology. Without loss of generality, we can assume that U ∈K. We have Y ∩ U ∈ U (otherwise, since U is an ultrafilter on Y, Y \ U ∈ U , so X \ U ∈ K Y,U , and thus in particular z ∈ X \ U : a contradiction), hence U ∈ K Y,U and x ∈ U , in particular. Then, z ∈ Cl({x}).
(2) Let U be an ultrafilter on Y and let Ω be an open neighborhood of x U , with respect to the constructible topology. Since the collection of all clopen sets of X is a basis for the open sets of X cons , we can assume, without loss of generality, that Ω = U ∩ (X \ V ), for some U, V ∈K. It follows immediately that U ∩ Y and Y \ V belong to U (otherwise, either X \ U or V would belong to K Y,U , then we would have a contradiction since K Y,U = {x U } and x U ∈ U ∩ (X \ V )). Thus, Ω ∩ Y ∈ U and it is, in particular, nonempty. This proves that x U ∈ Cl cons (Y).
Conversely, let x ∈ Cl cons (Y). Note that the following collection of sets (subsets of Y): has the finite intersection property, sinceK is a collection of clopen sets of the compact space X cons . Pick an ultrafilter U on Y such that G ⊆ U (Lemma 2.1). We claim that x = x U . To see this, since X is a T 0 space, it suffices to show that x and x U have the same set of open neighborhoods in X (with respect to the given (spectral) topology). Let U be an open and quasi-compact neighborhood of x. It follows Y ∩ U ∈ G ⊆ U . Thus U ∈ K Y,U and, in particular, x U ∈ U . Conversely, assume, by contradiction, that there is an open and compact neighborhood U of We apply the previous result to the prime spectrum of a ring.

2.4.
Corollary. Let R be a ring, X := Spec(R) (equipped with the Zariski topology), Y a subset of X and U an ultrafilter on Y . Then, is a prime ideal which coincides with the ultrafilter limit point x Y,U of X defined in Proposition 2.3.
Proof. By an argument similar to that used in [1, Lemma 2.4] (see also [13]) it can be easily shown that P U is a prime ideal of R.
In the other case, we have V (I) ∩ Y ∈ U and, since I is finitely generated. P Y,U ∈ V (I) = H. Recalling that K Y,U is a singleton, the conclusion follows immediately.

2.5.
Remark. Let X be a spectral space and Y be a subset of X . We say that Y is ultrafilter closed in X if x Y,U ∈ Y, for any ultrafilter U on Y. By Proposition 2.3, it follows that the collection of all the subsets of X that are ultrafilter closed is the family of closed sets for a topology X , that we call the ultrafilter topology of the spectral space X . If we denote by X ultra the space X endowed with the ultrafilter topology, then by Proposition 2.3 we have X ultra = X cons . Therefore, from Proposition 2.3 and Corollary 2.4, when X is the prime spectrum of a commutative ring, we reobtain as a particular case [13, Theorem 8].
2.6. Proposition. Let X be a spectral space and Y be a quasi-compact subspace of X . Then, the generic closure Y gen of Y in X is closed in X cons . Proof. Preserve the notation of Proposition 2.3, and let U be an ultrafilter on Y gen . It is sufficient to show that x U := x Y gen ,U ∈ Y gen . If not, for each y ∈ Y, there is an open and compact open neighborhood Ω y of y such that x U / ∈ Ω y . By compactness, the open cover {Ω y | y ∈ Y} of Y in X cons has a finite subcover, for some i ∈ {1, 2, . . ., n}. Thus, by Proposition 2.3(1), we have x U ∈ Ω yi , a contradiction.

The Kronecker function ring (after Halter-Koch) and the Zariski-Riemann surface
Let K be a field and let A be any subring of K. Denote by Zar(K|A) the set of all the valuation domains having K as quotient field and containing A as a subring.
As is well known, Zariski [31] (or, [32, Volume II, Chapter VI, §1, page 110]) introduced and studied the set Z := Zar(K|A) together with a topological structure defined by taking, as a basis for the open sets, the subsets This topology is called the Zariski topology on Z = Zar(K|A) and Z, equipped with this topology, denoted also later by Z zar , is usually called the (abstract) Zariski-Riemann surface of K over A.
On the set Z = Zar(K|A) we can also consider the constructible topology, as defined in the previous section, and we denote, as usual, Z cons the space Z endowed with the constructible topology.
In this section, we show that both Zar(K|A) zar and Zar(K|A) cons are spectral spaces, by giving in this general setting the explicit construction of a ring whose prime spectrum, equipped with the Zariski topology (respectively, constructible topology), is homeomorphic to Zar(K|A) zar (respectively, Zar(K|A) const ).
Let K be a field and T an indeterminate over K. For every W ∈ Zar(K(T )), it is well known that V := W ∩ K ∈ Zar(K) [15,Theorem 19.16(a)] and conversely, for each V ∈ Zar(K), there are infinitely many valuation domains 3.1. Proposition. Let K be a field and T an indeterminate over K.
(1) The canonical map ϕ : Proof. (1) The map ϕ is clearly surjective by the previous remarks. It is also a continuous map since, for each finite subset F of K and for each basic open set B (2) It is obvious that ϕ| Zar0(K(T )) : Zar 0 (K(T )) zar → Zar(K) zar is a bijection and, by (1), is a continuous map. The conclusion will follows if we show that the map ϕ| Zar 0 (K(T )) is also open. Let h ∈ K(T )\{0}, say with a i and b j in K, for i = 0, 1, . . ., r and j = 0, 1, . . ., s. Let V (T ) be a valutation domain in Zar 0 (K(T )), let v be the valuation on K defining V and let v * be the valuation associated to V (T ), i.e., v * (T ) = v(1) = 0 and, for each nonzero Now, for all i ∈ {0, 1, . . ., r} and j ∈ {0, 1, . . ., s} such that both a i and b j are nonzero, set: Then, it is not hard to verify that ϕ(B (see also the proof of [4, Lemma 1]), hence the continuous bijective map ϕ| Zar0(K(T )) is open, and so a homeomorphism. Now recall the following key notion introduced by Halter-Koch [18, Definition 2.1], providing an axiomatic approach to the theory of Kronecker function rings.
Let K be field, T an indeterminate over K, and R a subring of K(T ). We call R a K-function ring (after Halter-Koch) if T and T −1 belong to R and, for each We collect in the next proposition several properties of K-function rings that will be useful in the following.
3.2. Proposition. Let K be a field, T an indeterminate over K and let R be a subring of K(T ). Assume that R is a K-function ring.
3.4. Proposition. Let K be a field, T an indeterminate over K, and R a subring of K(T ). Then, the following conditions are equivalent.
(i) R is a K−function ring.
As a consequence of Propositions 3.1(2), 3.9 and 3.3, we deduce immediately the following.
3.5. Corollary. Let K be a field, T an indeterminate over K and R (⊆ K(T )) a K-function ring. Set A R := R ∩ K. Then, the canonical map ϕ : Zar(K(T )|R) −→ Zar(K|A R ), W → W ∩ K, is a topological embedding, with respect to the Zariski topology.
As an application of the previous corollary we reobtain in particular [19, Corollary 2.2, Proposition 2.7 and Corollary 2.9]. More precisely, 3.6. Corollary. Let K be a field, A any subring of K and T an indeterminate over K. Then,

3.7.
Remark. Note that the noteworthy progress provided by Corollary 3.6 concerns the case where A is a proper subfield of K. As a matter of fact, if A is an integrally closed domain and K is its quotient field, statements (2) and (3)  3.8. Corollary. Let K be a field and A a subring of K. If Y is a nonempty subset of Zar(K|A) (equipped with the Zariski topology) and U is an ultrafilter on Y , then 3.9. Proposition. We preserve the notation of Proposition 3.1 and, now, let Zar(K(T )) and Zar(K) be endowed with the constructible topology. Then, the canonical (surjective) map ϕ : Zar(K(T )) cons → Zar(K) cons is continuous and (hence) closed. In particular, ϕ| Zar 0 (K(T )) is a homeomorphism of Zar 0 (K(T )) cons onto Zar(K) cons .
A "constructible" version of Corollary 3.6(2 and 3) can also be easily deduced from the previous considerations.
3.11. Corollary. Let K be a field, A any subring of K, T an indeterminate over K, and let Kr(K|A) be as in Corollary 3.6.
(1) The canonical map ϕ : Zar(K(T )|Kr(K|A)) cons → Zar(K|A) cons , defined by W → W ∩ K, is a homeomorphism. (2) The canonical map ψ : Spec(Kr(K|A))) cons → Zar(K|A) cons , defined by Q → Kr(K|A) Q ∩ K, is a homeomorphism. In particular, Zar(K|A) cons is a spectral space canonically homeomorphic to the prime spectrum of the absolutely flat ring canonically associated to the K-function ring Kr(K|A). Proof.
(1) As observed in the proof of Corollary 3.6(1), Kr(K|A) is a K-function ring, hence this statement follows from Proposition 3.9.

Some Applications
Let K be a field, and let A be a subring of K. In this section, we use constantly that on the space Zar(K|A) the contructible topology coincides with the ultrafilter topology (Remark 2.5) and we give some applications of the results of the previous sections to the representations of integrally closed domains as intersections of valuation overrings. 4.1. Proposition. Let K be a field, A be a subring of K and U be a subset of Z := Zar(K|A). Let Y ′ and Y ′′ be two subsets of a given subset U of Z and assume that their closures in U , with the subspace topology induced by the constructible topology of Z, coincide, i.e., Cl cons (Y ′ Proof. Assume, by contradiction, that there is an element

4.2.
Remark. Note that the previous proposition is stated very generally using a "relative-type" formulation. However, it is clear that, if we take any two subsets Y ′ and Y ′′ of Z, the rôle of U can be played by any subset of Z (including Z) Let Σ be a collection of subrings of a field K, having K as quotient field. We say that Σ is locally finite if each nonzero element of K is noninvertible in at most finitely many of the rings belonging to Σ.
The following easy result will provide a class of integral domains for which the equality Proof. By Proposition 2.3(2) and [7,Remark 3.2], it is enough to show that K = A U (= {x ∈ K | B x ∩ Σ ∈ U }), for every nontrivial ultrafilter U on Σ. By contradiction, assume that there exists an element x 0 ∈ K \A U . Then Σ\B x0 ∈ U , and so it is infinite, since U is nontrivial (an ultrafilter containing a finite set is trivial). This implies that x 0 is noninvertibile in infinitely many valuation domains belonging to Σ, a contradiction.
As a consequence of the previous lemma, we have that, if an integral domain admits two distinct infinite and locally finite representations as intersection of valutation domains, then the converse of Proposition 4.1 does not hold. An explicit example is given next.
Our goal is to represent A as a locally finite intersection of valuation domains in two different ways. In fact, we can use one description to generate an infinite number of different such representations.
First, note that B can be represented as an intersection of DVR's which are obtained by localizing at its height-one primes, i.e., B = {B P | P ∈ Spec(B), ht(P ) = 1}. It is well known that this collection is locally finite. Now, note that A is a local domain with maximal ideal T2) . Choose any valuation overring W of A such that M W (the maximal ideal of W ) is generated by T 3 and lies over the maximal ideal of A. It is easy to see that many such valuation domains of the field k(T 1 , T 2 , T 3 ) exist (e.g., let W ′ be a valuation overring of B with maximal ideal M ′ lying over M B and such that the residue field W ′ /M ′ is canonically isomorphic to k(T 3 ), which is the residue field B/M B , then the domain V + M ′ , with V as in the previous paragraph, can serve as the desired domain W ). Now, the intersection R := {B P | P ∈ Spec(B), ht(P ) = 1} W is clearly a locally finite intersection. We claim that any choice, as above, of the domain W will yield R = A.
To prove our claim, we note first that it is obvious that R is an overring of A. So, we need to prove that R ⊆ A. Observe that the ideal M B is an ideal of A as well as of B. It follows easily that M B is a prime ideal of R, since R ⊂ B. Then, given an element r ∈ R, we can write r = ψ + f , where ψ ∈ k(T 3 ) and f ∈ M B . However, f ∈ M B ⊆ W and so ψ ∈ W . It is clear though that W ∩ k(T 3 ) = V . It follows that ψ ∈ V and so r ∈ A. Hence, we have proven that R ⊆ A.
The following Proposition is the key step in proving the main results of the section.
Since, by what we observed above, γ −1 (Max(Kr(C))) ⊆ C 0 , then V ′ (T ) ∈ C 0 . 4.7. Remark. Note that, with the notation and assumptions of Theorem 4.6, As a matter of fact, { W ∈ Zar(K(T )) | W ⊇ V (T ), for some V ∈ C} = { W ∈ Zar(K(T )|Kr(C)) | W ⊇ V (T ), for some V ∈ C}. By Proposition 3.3, we have Zar(K(T )|Kr(C)) = Zar 0 (K(T )|Kr(C)), thus We recall some properties of semistar operations. Let A be an integral domain and let K be the quotient field of A. We denote by F (A) the set of all the nonzero A-submodules of K, and by f (A) the set of all the nonzero finitely generated A- is called a semistar operation on A if, for each 0 = x ∈ K and for all E, F ∈ F (A), the following properties hold: A semistar operation of finite type ⋆ on A is a semistar operation such that, for every E ∈ F (A), An e.a.b. semistar operation ⋆ on A is a semistar operation such that, for all ( Proof. Let T be an indeterminate over K. By [10, Remark 3.5(b)], it is enough to show that condition (ii) is equivalent to the following . Assume that the equality Cl cons (Y ′ ) ↑ = Cl cons (Y ′′ ) ↑ holds. Keeping in mind the notation introduced before Theorem 4.6 and applying Corollary 3.11 (1), it follows easily that, inside Zar(K(T )), Cl cons (Y ′ 0 ) ↑ = Cl cons (Y ′′ 0 ) ↑ . By using Proposition 4.1 and Remark 4.8, we have The conclusion follows immediately from Theorem 4.9.
From the previous theorem, we deduce immediately the following 4.11. Corollary. Let A be an integrally closed domain. If each representation of A is dense in Zar(K|A) cons , then A is a vacant domain.
(v)⇒ (iii). Take a set Y ′′ as stated in (v). Then, (iii) follows immediately from Proposition 2.6 and Remark 4.8, by taking Y := Y ′′↑ . 4.14. Corollary. Let A be an integral domain and K its quotient field. Let Y be a subset of Zar(K|A) and set Y := Cl cons (Y ) ↑ . Then, (∧ Y ) f = ∧ Y = ∧ Cl cons (Y ) .
Proof. In view of Proposition 2.6, Y is closed, with respect to the constructible topology. Thus ∧ Y is of finite type, by Theorem 4.13, and hence the equality (∧ Y ) f = ∧ Y follows immediately by Theorem 4.9, since Cl cons (Y ) ↑ = Y ↑ (= Cl cons ( Y ↑ )). Moreover, the semistar operation ∧ Cl cons (Y ) is of finite type, by Theorem 4.13, and thus the last equality follows by applying Theorem 4.9.
The next example illustrates the possibility that the sets Y, Y ′ and Y ′′ in Theorem 4.13 can form a proper chain of sets.

4.15.
Example. Let k be a field and T 1 , T 2 two indeterminates over k. Let A be the two-dimensional, integrally closed, local domain k[T 1 , T 2 ] (T1,T2) with quotient field K := k(T 1 , T 2 ). Let ⋆ be the b-operation on A. It is well known that the b-operation is an e.a.b. operation of finite type. Hence, it satisfies the equivalent conditions of Theorem 4.13. Our goal is to show that there is a great deal of flexibility in the choice of the sets Y, Y ′ and Y ′′ in the theorem. First, note that if the valuation domains in Zar(K|A) are ordered by inclusion then any chain is finite [15,Corollary 30.10] and, hence, obviously there are minimal elements. Any such minimal valuation overring V will necessarily have maximal ideal M V lying over the maximal ideal (T 1 , T 2 ) of A. The standard definition of the boperation involves extending an ideal (or, more generally a sub-A-module of K) to all valuation overrings. It is clearly sufficient to extend to just those valuation overrings that are minimal. So, any subcollection of Zar(K|A) which contains all the minimal elements will generated the b-operation. It is not clear that the collection of minimal valuation overrings is closed under the Zariski or the constructible topology.
• Consider the members of Zar(K|A) which do not contain the elements 1 T1 , 1 T2 . This is a closed, quasi-compact subset of Zar(K|A) zar . It can also be thought of as being those valuation domains in Zar(K|A) whose maximal ideal dominates (T 1 , T 2 ) in A. Hence, it contains the minimal valuation overrings and is sufficient to generate the b operation. We can let this collection be denoted by Y ′′ in Theorem 4.14.
• The set Y ′′ , described above, is a (proper) closed subset of Zar(K|A) zar .
Hence, it is also closed in Zar(K|A) cons . Moreover, any closed subset of Zar(K|A) cons is compact. Hence, to obtain our set Y ′ , we can choose any closed subset of Zar(K|A) cons which contains Y ′′ . Since any single point is closed in Zar(K|A) cons , we can let Y ′ be the union of Y ′′ and any other single valuation overring, for example, the localization of A at a height-one prime. • The set Y should contain all overrings of its members. An obvious choice then is to let Y be all of Zar(K|A) cons . Since this is the entire space it is trivially closed (in Zar(K|A) cons ) and generates the b-operation.
This then gives three different sets Y ′′ ⊂ Y ′ ⊂ Y with the notation of Theorem 4.13, all associated with the same (semi)star operation. By using Remark 2.2, we can restate Corollaries 4.10 and 4.14 as follows: 4.16. Corollary. Let A be an integrally closed domains and K be its quotient field. Then the following conditions are equivalent.
(i) A is a vacant domain.
(ii) Each representation of A is dense in Zar(K|A), with respect to the inverse topology.
4.17. Corollary. Let A be an integral domain and K its quotient field. Let Y be a subset of Zar(K|A). Then, (∧ Y ) f = ∧ Cl inv (Y ) .