KARIM BELAID, OTHMAN ECHI, AND RIYADH GARGOURI Received 15 December 2004 and in revised form 6 July 2005
We deal with two classes of locally compact sober spaces, namely, the class of locally spec- tral coherent spaces and the class of spaces in which every point has a closed spectral neighborhood (CSN-spaces, for short). We prove that locally spectral coherent spaces are precisely the coherent sober spaces with a basis of compact open sets. We also prove that CSN-spaces are exactly the locally spectral coherent spaces in which every compact open set has a compact closure.
1. Introduction
Asoberspace is aT0-space in which every closed irreducible subset is a singleton closure.
Every sober spaceXcan be recovered from the frame (complete distributive Brouwerian lattice)ᏻ(X) of open sets as the prime spectrum Spec(ᏻ(X)) and eachT0-spaceY can be universally embedded into the sober space Spec(ᏻ(Y)).
A space will be calledcompact if every open cover has a finite subcover, andlocally compact if every neighborhood of a point contains a compact neighborhood. The the- ory of locally compact sober spaces is completely captured by the theory of continuous frames (continuous distributive lattices), because for each locally compact sober space X, the latticeᏻ(X) is a continuous frame such thatX∼=Spec(ᏻ(X)) and every continu- ous frameLhas a locally compact sober space Spec(L) as its prime spectrum such that L∼=ᏻ(Spec(L)). A locally compact sober space has a basis of compact open sets if and only if its frameᏻ(X) of open sets is algebraic [1, page 115], and the spectrum of every algebraic frame is a locally compact sober space with a basis of compact open sets. A space Xwhose frameᏻ(X) is algebraic is sometimes called a quasi-Boolean space [4].
LetXbe a topological space andAa subset ofX. ThesaturationofAis Sat(A)= ∩
U|A⊆UandUis open inX. (1.1) One says thatAissaturated, ifA=Sat(A). InT0-spaces the right classes of compact sub- sets are the saturated ones. A subset is compact if and only if its saturation is compact.
A space is calledcoherent if the intersection of two saturated compact sets is compact.
Copyright©2005 Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences 2005:15 (2005) 2421–2427 DOI:10.1155/IJMMS.2005.2421
A space is calledstably compactif it is compact, locally compact, coherent, and sober. One of the principal results of the theory of stably compact spaces [2, page 474] is that aT0- space is stably compact if and only if the patch topology is a compact Hausdorfftopology.
Thepatch topologyis an invention of Hochster; a basis of its open sets consists of the given open sets plus the complements of the compact saturated sets. Stably compact spaces and partially ordered compact Hausdorffspaces are practically one and the same thing [2, page 482]. The frameᏻ(X) of a stably compact space is a stably continuous frame (i.e., has a multiplicative way-below relation) with compact top element; conversely, a stably continuous frame with compact maximal element has a stably compact spectrum. Thus stably compact spaces are again practically the same as stably continuous frames with compact top element [2, page 489].
IfRis a commutative ring with an identity, then its prime ideal spectrum is the prime spectrum of the latticeLof radical ideals and that is an algebraic frame and thus Spec(R) is a locally compact sober space with a basis of compact open sets (and due to the presence of an identity in R) it is compact. It is Hochster’s merit to have shown [3] that every compact, locally compact, coherent, sober space with a basis of compact open sets arises in this way. One says that a space isspectral, if it is a compact, locally compact, coherent, sober space with a basis of compact open sets [3].
The primitive spectrum of aC∗-algebra is a locally compact Baire space (although if the algebra is not separable, it may fail to be sober); so locally compact spaces receive a considerable interest in the operator algebra community.
Due to the fact that authors in theoretical computer science dealing with the denota- tional semantics of programming languages have been interested inT0 spaces and con- tinuous lattices since Scott’s article [5], there is considerable interest in this topic.
The role of compact, locally compact, and sober spaces with a basis of compact open sets in Stone’s theorem on the representation of distributive lattices and similar contexts was discussed in [4] in 1972.
Hochster [3] has called a spacelocally spectralif it has a cover by open spectral sub- spaces.
Here we are interested in the study of two classes of locally compact sober spaces, namely, the class of locally spectral coherent spaces and the class of spaces in which every point has a closed spectral neighborhood (CSN-spaces, for short).
This paper contributes two “local-global” results, the first one explaining that locally spectral coherent spaces are precisely the coherent sober spaces with a basis of compact open sets, and the second that CSN-spaces are exactly the locally spectral coherent spaces in which every compact open set has a compact closure.
2. Locally spectral coherent spaces
This section deals with an intrinsic topological characterization of locally spectral coher- ent spaces.
First, recall some characterizations M. Hochster.
Theorem 2.1 [3]. The locally spectral spaces are precisely the underlying spaces of pre- schemes.
Theorem2.2 [3]. The following conditions on a topological spaceXare equivalent:
(i)Xis a coherent locally spectral;
(ii)Xis the underlying space of an open subscheme of an affine scheme;
(iii)Xis the underlying space of some scheme;
(iv)Xis homeomorphic with an open subspace of a spectral space.
First, we need two lemmas, the first one is cited in [1, Proposition 7, page 122].
Lemma2.3. LetXbe a topological space andUa nonempty open set ofX. Then the mapping V →V defines a bijection from the set of irreducible nonempty closed subsets ofU onto the set of irreducible nonempty closed subsets ofX meetingU. The inverse bijection isZ→ Z∩U.
The second lemma is a special case of [2, Exercise O-5.15(ii)].
Lemma2.4. LetXbe a sober space andUa nonempty open set ofX. ThenUis sober.
Theorem2.5. The following conditions on a topological spaceXare equivalent:
(i)Xis a locally spectral coherent space;
(ii)there exists a spectral spaceYwith a unique closed pointωsuch thatXis homeomor- phic with the subspaceY\ {ω}ofY;
(iii)Xis a coherent sober space with a basis of compact open sets;
(iv)Xis an open subset of a spectral space.
Proof. As (i)⇔(iv) according toTheorem 2.2, it suffices to prove the equivalence between conditions (ii), (iii), and (iv).
(ii)⇒(iii). Let (Y,᐀) be a spectral space with a unique closed pointω.
(a) SinceY is the unique open set ofY containingω, the topology ofX is{U∈᐀| ω /∈U}. HenceXis coherent with a basis of compact open sets.
(b) ByLemma 2.4, the open subsetXof the sober spaceYis also sober.
(iii)⇒(ii) and (iv).
(c) Letω /∈XandX=X∪ {ω}. Consider the topology᐀ =᐀∪ {X}onX. It is easily seen that (X, ᐀) is a compact coherent T0-space with a basis of compact open sets.
(d) LetCbe a nonempty irreducible closed subset of (X, ᐀). Then ω∈CandC= K∪ {ω}, whereKis a closed subset of (X,᐀). We discuss two cases.
Case 1. K= ∅. In this case,C= {ω}has a generic point.
Case 2. K= ∅. NecessarilyKis an irreducible closed subset of (X,᐀). Hence there exists x∈Xsuch thatC= {x}X= {x}X.
Therefore,Xis a spectral space with a unique closed pointω(ii) and X is an open subspace (iv).
(iv)⇒(iii). Applying Lemma 2.4,X is sober. Let U be an open set of a topological spaceX. IfXhas a basisᏮof compact open sets closed under finite intersections, then ᏮU= {O∈Ꮾ|O⊆U}is a basis of compact open sets ofUwhich is closed under fi- nite intersections, proving thatXis a coherent sober space with a basis of compact open
sets.
3. CSN-spaces
We divide the proof of our main result into a sequence of lemmas.
Lemma3.1. LetXbe a CSN-space. ThenXis coherent and has a basis of compact open sets.
Proof. (1) Letx∈X. SinceX is a CSN-space, there exists an open setOxand a closed spectral subsetCx ofX such thatx∈Ox⊆Cx. LetᏮx be a basis of compact open sets ofCxandᏮᏻx= {O∈Ꮾx|O⊆Ox}. The elements ofᏮᏻxare compact open sets ofX.
ThusᏮ=
x∈XᏮᏻxis a basis of compact open sets ofX.
(2) To prove thatXis coherent, it suffices to show that the intersection of two elements ofᏮis compact. LetO1,O2 be two elements ofᏮ. Then there existx,y∈X such that O1∈Ꮾᏻx andO2∈Ꮾᏻy. SinceO1andO2∩Cxare two compact open sets ofCx,O1∩
O2=O1∩(O2∩Cx) is compact.
Lemma3.2. A spaceXin which every point has a closed sober neighborhood is sober.
Proof. LetCbe a nonempty irreducible closed subset ofX andx∈C. There exists an open setOx and a closed sober subset Cx ofX such that x∈Ox⊆Cx. Of course,C= (Ox∩C)∪((X−Ox)∩C) yieldsC=Ox∩C⊆Cx, by irreducibility ofC. HenceCis an irreducible closed subset of the sober spaceCx. Therefore,Chas a generic point.
Since spectral spaces are precisely the coherent compact sober spaces with a base of compact open sets, the following result is an immediate consequence of Lemmas3.1and 3.2.
Corollary3.3. LetXbe a topological space. Then the following statements are equivalent:
(i)Xis a spectral space;
(ii)Xis a compact CSN-space.
LetXandYbe two topological spaces and f :X→Y a continuous map. Hochster [3]
has called f spectralif for each compact open setUofY, f−1(U) is compact inX.
We need to recall the patch topology [3]. LetX be a topological space. By thepatch topologyonX, we mean the topology which has as a subbasis for its closed sets the closed sets and compact open sets of the original space. IfX has a basis of compact open sets which is closed under finite intersections, then the patch topology has the compact open sets and their complements as an open subbasis. By apatchwe mean a closed set in the patch topology. Recall that the patch topology associated to a spectral space is Hausdorff and compact [3]. The setXequipped with the patch topology will be denoted byXpatch.
The following remarks are direct consequences of Hochster’s results [3].
Remarks 3.4. (1) LetX be a topological space which has a basis of compact open sets closed under finite intersections. Then the following statements are equivalent:
(i)Xis spectral;
(ii)Xpatchis compact.
(2) LetXbe a spectral space andYa subspace ofX. ThenY is spectral if and only ifY is a patch inX.
(3) IfYis a patch in a spectral spaceX, thenY=
[{y} |y∈Y].
Lemma3.5. LetXbe a spectral space andYa topological space which has a basis of compact open sets closed under finite intersections. Let f :X→Y be a continuous map. Then the following statements are equivalent:
(i) f is a spectral map;
(ii) f :Xpatch→Ypatchis continuous.
Proof. (i)⇒(ii). For each compact open setU of Y, f−1(U) is a compact open set of X and f−1(Y−U)=X− f−1(U) is a complement of a compact open set ofX. Thus
f :Xpatch→Ypatchis continuous.
(ii)⇒(i). LetUbe a compact open set ofY. ThenUis closed inYpatch. Hence f−1(U) is a closed subset of the compact spaceXpatch. Thus f−1(U) is compact.
Corollary3.6. The patch topology of a closed spectral subset of a CSN-spaceXcoincides with the topology induced by the patch topology ofX.
Proof. LetCbe a closed spectral subset ofXand᐀patchthe topology induced by the patch topology ofXonC. SinceCis a closed spectral subset ofX, the canonical injectioni:C Xis a spectral map. Hence usingLemma 3.5, 1C:Cpatch→(C,᐀patch) is continuous. Now, sinceCpatchis a compact space and (C,᐀patch) is Hausdorff, 1Cis a homeomorphism.
Lemma3.7. LetXbe a CSN-space. Then each closed compact subset ofXis a spectral sub- space ofX.
Proof. LetCbe a compact closed subset ofX.
(a) Clearly,Cis aT0-space, sinceXisT0.
(b) LetᏮbe a basis of compact open sets ofXwhich is closed under finite intersec- tions. Of course,ᏮC= {U∩C|U∈Ꮾ}is a basis of compact open sets of Cwhich is closed under finite intersections.
(c) Finally, one may easily check that any closed subset of a sober space is sober.
Therefore,Cis a spectral space.
Corollary3.8. LetXbe a CSN-space. Then for each compact open setOofX, there exists a closed spectral subspaceCofXsuch thatO⊆C.
Proof. For eachx∈O, there exists a closed spectral neighborhoodCx ofx. HenceO⊆ [Cx|x∈U]. SinceO is compact, there exists a finite subsetM of U such thatO⊆ [Cx|x∈M]. LetC=
[Cx|x∈M]. ThenCis a closed compact subspace ofX. Ac-
cording toLemma 3.7,Cis a spectral subspace ofX.
Lemma3.9. LetY be a nonempty subset of a CSN-spaceX. Then the following statements are equivalent:
(i)Yis a patch inX;
(ii)Yis a CSN-subspace ofXand for each compact open setOofX,O∩Y is compact.
Proof. (ii)⇒(i). Letx∈YXpatchandOxbe a compact open set ofXsuch thatx∈Ox. Then Ox∩Y is a nonempty compact open set ofY. According toCorollary 3.8, there exists a closed spectral subspaceCxofYsuch thatOx∩Y⊆Cx.
Now, remark that the clopen subsets ofXpatchform a basis of the patch topology ofX.
LetᏴ= {(V∩Y)∩Cx|x∈VandVis clopen inXpatch}. The intersection of any fi- nite family of elements ofᏴis of the form (U∩Y)∩Cx, whereUis clopen inXpatchand x∈U. Now,x∈YXpatchimplies that (Ox∩Y)∩(U∩Y)= ∅so that (U∩Y)∩Cx= ∅. HenceᏴhas the finite intersection property.
On the one hand, sinceCxis compact inYpatch, we have[W|W∈Ᏼ]= ∅. On the other hand,[V |x∈V andV is clopen inXpatch]= {x}, since Xpatchis aT2-space. It follows thatx∈Y, and thusYXpatch=Y. Therefore,Yis a patch inX.
(i)⇒(ii). Letx∈Y. SinceX is a CSN-space, there exists a closed spectral neighbor- hoodCxofx. ByCorollary 3.6,Cx∩Y is a patch inCx. HenceCx∩Yis a closed spectral neighborhood ofxinY. ThusY is a CSN-space.
LetObe a compact open set ofX. UsingCorollary 3.8, there exists a closed spectral subspaceCofX such thatO⊆C. SinceC∩Y is a patch inC,C∩Y is a spectral space.
According toCorollary 3.6,O∩Y =(O∩Y)∩Cis a patch inC. It follows thatO∩Y is
a compact open set ofY.
Corollary3.10. LetXbe a CSN-space and letY⊆X. Then the following properties hold.
(1)IfUis a compact open set ofX, thenUandUare spectral subspaces ofX.
(2)IfY is a compact subset ofX, thenYis a spectral space.
(3)Yis spectral if and only ifYis a patch compact subset ofX.
Proof. (1) LetUbe a compact open set ofX; byCorollary 3.8, there exists a closed spectral subsetFofXsuch thatU⊂F. Hence, asUis a closed subspace of the spectral spaceF, it is a spectral subspace ofX. Now,Uis a patch inU, so thatUis spectral.
(2) SinceYis a compact subset ofX, there exists a compact open setUofXsuch that Y⊆U. HenceY is a patch in the spectral spaceUby (1). It follows thatY is a spectral space [3].
(3) Suppose thatY is spectral. ThenYis compact. HenceY is spectral by (2). Accord- ing to M. Hochster, a spectral subspace of a spectral space is necessarily a patch. ThusY is a patch inY. ByCorollary 3.6,Yis a patch inX.
Conversely, suppose thatYis a compact patch subset ofX. ThenYis a CSN-subspace ofXbyLemma 3.9. NowCorollary 3.3permits to conclude thatYis spectral.
The following is an immediate consequence of the above corollary.
Corollary3.11. LetX be a topological space with a basisᏮof compact open sets. Then the following statements are equivalent:
(1)Xis a CSN-space;
(2)for eachU∈Ꮾ,Uis a spectral subspace ofX.
Now, we are in a position to give a characterization of CSN-spaces.
Theorem3.12. LetXbe a topological space. Then the following statements are equivalent:
(1)Xis a CSN-space;
(2)Xhas the following properties:
(i)Xis a coherent sober space with a basis of compact open sets;
(ii)every compact open setUofXhas a compact closure (i.e.,Uis relatively compact).
Proof. (1)⇒(2). This follows immediately from Lemmas3.1and3.2andCorollary 3.10.
(2)⇒(1). Letx∈X. Then there exists a compact open set OofX such thatx∈O. Property (ii) forcesOto be a spectral subspace ofX. Therefore,Xis a CSN-space.
Corollary3.13. LetXbe a topological space. Then the following statements are equiva- lent:
(i)Xis a CSN-space;
(ii)Xis a locally spectral coherent space in which every compact open set has a compact closure.
Example 3.14. (1) It is easily seen that a spectral space is a CSN-space and a CSN-space is a locally spectral space (seeTheorem 3.12).
(2) A locally spectral space need not be a CSN-space. LetNbe the set of integers en- dowed with the Alexandroffdiscrete topology᐀of the natural order (here the open sets are∅andN, and↓x= {y∈N|y≤x}, wherex∈N). Clearly, (N,᐀) is a noncompact locally spectral space. But↓0 is a compact open set of (N,᐀) with closureN. Thus, ac- cording toTheorem 3.12, (N,᐀) is not a CSN-space.
(3) A CSN-space need not be spectral; the same example as in (2) does the job.
Acknowledgment
The authors gratefully acknowledge helpful comments and suggestions of the anonymous referee.
References
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[2] G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. W. Mislove, and D. S. Scott,Continuous Lattices and Domains, Encyclopedia of Mathematics and Its Applications, vol. 93, Cam- bridge University Press, Cambridge, 2003.
[3] M. Hochster,Prime ideal structure in commutative rings, Trans. Amer. Math. Soc.142(1969), 43–60.
[4] K. H. Hofmann and K. Keimel,A general character theory for partially ordered sets and lattices, Mem. Amer. Math. Soc.122(1971).
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97–136.
Karim Belaid: D´epartement des Math´ematiques, ´Ecole Sup´erieure des Sciences et Techniques de Tunis, 5 Avenue Taha Hussein, BP 56, Bab Mnara 1008, Tunisia
E-mail address:[email protected]
Othman Echi: Department of Mathematics, Faculty of Sciences of Tunis, University of Tunis El Manar, Campus Universitaire, 2092 El Manar II, Tunisia
E-mail address:[email protected]
Riyadh Gargouri: Department of Mathematics, Institute of Multimedia, Route Mharza Km 1, 5, BP 1030, 3018 Sfax, Tunisia
E-mail address:tn riadh [email protected]
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