Intrinsic characterizations of C -realcompact spaces
aa r X i v : . [ m a t h . GN ] J u l INTRINSIC CHARACTERIZATIONS OF C -REALCOMPACTSPACES SUDIP KUMAR ACHARYYA, RAKESH BHARATI, AND A. DEB RAY
Abstract. c -realcompact spaces are introduced by Karamzadeh and Keshtkarin Quaest. Math. 41(8), 2018, 1135-1167. We offer a characterization of thesespaces X via c -stable family of closed sets in X by showing that X is c -realcompact if and only if each c -stable family of closed sets in X with finiteintersection property has nonempty intersection. This last condition whichmakes sense for an arbitrary topological space can be taken as an alternativedefinition of a c -realcompact space. We show that each topological space canbe extended as a dense subspace to a c -realcompact space with some desiredextension property. An allied class of spaces viz CP -compact spaces akin tothat of c -realcompact spaces are introduced. The paper ends after examininghow far a known class of c -realcompact spaces could be realized as CP -compactfor appropriately chosen ideal P of closed sets in X . Introduction
In what follows X stands for a completely regular Hausdorff topological space. Asusual C ( X ) and C ∗ ( X ) denote respectively the ring of all real valued continuousfunctions on X and that of all bounded real valued continuous functions on X .Suppose C c ( X ) is the subring of C ( X ) containing those functions f for which f ( X )is a countable set and C ∗ c ( X ) = C c ( X ) ∩ C ∗ ( X ). Formal investigations of these tworings vis-a-vis the topological structure of X are being carried on only in the recenttimes. It turns out that there is an interplay between the topological structure of X and the ring and lattice structure of C c ( X ) and C ∗ c ( X ), which incidentally shedsmuch light on the topology of X . The articles [3], [4], [7], [8], [11] may be referred inthis context. The notion of c -realcompact spaces is the fruit of one such endeavoursin the study of X versus C c ( X ) or C ∗ c ( X ). A space X is declared c -realcompactin [8] if each real maximal ideal M in C c ( X ) is fixed in the sense that there exitsa point x ∈ X such that for each f ∈ M , f ( x ) = 0. M is called real when theresidue class field C c ( X ) /M is isomorphic to the field R . A number of interestingfacts concerning these spaces is discovered in [8]. These may be called countableanalogues of the corresponding properties of real compact spaces as developed in [6],chapter 8. In the present article we offer a new characterization of c -realcompactspaces on using the notion, c -stable family of closed sets in X . A family F ofsubsets of X is called c -stable if given f ∈ C ( X, Z ), there exists F ∈ F such that f is bounded on F . Mathematics Subject Classification.
Key words and phrases. c -realcompact spaces, Banaschewski compactification, c -stable familyof closed sets, ideals of closed sets, initially θ -compact spaces.The second author acknowledges financial support from University Grand Commission, NewDelhi, for the award of research fellowship (F. No. 16-9(June 2018)/2019 (NET/CSIR)). We define a topological space X (not necessary completely regular ) to be c c -realcompact if each c -stable family of closed sets in X with finite intersec-tion property has nonempty intersection. We check that this new notion of c c -realcompactness agrees with the already introduced notion of c -realcompactness in[8], within the class of zero-dimensional Hausdorff spaces (Theorem 2.3). We re-establish a modified version of a few known properties of c -realcompact spaces usingour new definition c c -raelcompactness (Theorem 2.4). Furthermore we realize thatany topological space X can be extended as a dense subspace to a c c -realcompactspace υ X enjoying some desired extension properties (Theorem 2.5). While con-structing this extension of X , we follow closely the technique adopted in [9]. Theresults mentioned above constitute the first technical section viz § P of closed sets in X is called an ideal of closed sets if A ∈ P , B ∈ P and C is a closed subset of A imply that A ∪ B ∈ P and C ∈ P . Let Ω( X )stand for the aggregate of all ideals of closed sets in X . For any P ∈ Ω( X ) let C P ( X ) = { f ∈ C ( X ) : cl X ( X \ Z ( f )) ∈ P} , here Z ( f ) = { x ∈ X : f ( x ) = 0 } is thezero set of f in X . It is well known that C P ( X ) is an ideal in the ring C ( X ), see [1]and [2] for more information on these ideals. With referance to any such P ∈ Ω( X ),we call a family F of subsets of X c P -stable if given f ∈ C ( X, Z ) ∩ C P ( X ) thereexists F ∈ F such that f is bounded on F . We define a space X to be c P -compactif any c P -stable family of closed sets in X with finite intersection property has non-empty intersection. It is clear that a zero-dimensional space X is c c -realcompact ifit is already c P -compact.We have shown that if X is a noncompact zero-dimensional space and P ∈ Ω( X )such that X is c P -compact, then there exists an R ∈ Ω( X ) such that R & P and X is c R -compact. Thus within the class of zero-dimensional noncompact spaces X ,there is no minimal member P ∈ Ω( X ) in the set inclusion sense of the term forwhich X becomes c P -compact (Theorem 3.3). In the concluding portion of § c -realcompact spacescould be achieved as c P -compact spaces for appropriately chosen P ∈ Ω( X ). Forany infinite cardinal number θ , X is called finally θ -compact if each open coverof X has a subcover with cardinality < θ (see [10]). In this terminology finally ω -compact spaces are Lindel¨of and finally ω -compact spaces are compact. It isrealized that a c -realcompact space X is finally θ -compact if and only if it is c Q -compact, where Q is the ideal of all closed finally θ -compact subsets of X (Theorem3.5). A special case of this result reads: X is Lindel¨of when and only when X is c α -compact where α is the ideal of all closed Lindel¨of subsets of X .2. Properties of c c -realcompact spaces and c c -realcompactifications Before stating the first technical result of this section, we need to recall a fewterminologies and results from [4] and [8]. Our intention is to make the presentarticle self contained as far as possible. An element α on a totally ordered field F is called infinitely large if α > n for each n ∈ N . It is clear that F is archimedean ifand only if it does not contain any infinitely large element. If M is maximal idealin C c ( X ) then the residue class field C c ( X ) /M is totally ordered according to thefollowing definition: for f ∈ C ( X ) M ( f ) ≧ g ∈ M suchthat f ≧ Z ( g ). Here M ( f ) stands for the residue class in C ( X ) /M , whichcontains the function f . NTRINSIC CHARACTERIZATIONS OF C -REALCOMPACT SPACES 3 Theorem 2.1. ( Proposition 2.3 in [8])
For a maximal ideal M in C c ( X ) and for f ∈ C c ( X ) , | M ( f ) | is infinitely large in C c ( X ) /M if and only if f is unbounded onevery zero set of Z c ( M ) = { Z ( g ) : g ∈ M } . It is proved in [4], Remark 3.6 that if X is a zero-dimensional space, then the setof all maximal ideals of C c ( X ) equipped with hull-kernel topology, also called thestructure space of C c ( X ) is homeomorphic to the Banaschewski compactification β X of X . Thus the maximal ideals of C c ( X ) can be indexed by virtue of the pointsof β X . Indeed a complete description of all these maximal ideals is given by thelist { M pc : p ∈ β X } , where M pc = { f ∈ C c ( X ) : p ∈ cl β X Z ( f ) } with M pc is a fixedmaximal ideal if and only if p ∈ X (see Theorem 4.2 in [4]). It is well known thatany continuous map f : X → Y , where X and Y are both zero-dimensional spaceswith Y compact also, has an extension to a continuous map ¯ f : β X → Y (we callthis property, the C -extension property of β X ) (see Remark 3.6 in [4]). It followsthat for a zero-dimensional space X , any continuous map f : X → Z (also writtenas f ∈ C ( X, Z )), has an extension to a continuous map f ∗ : β X → Z ∗ = Z ∪ { ω } ,the one point compactification of Z . We also write f ∗ ∈ C ( β X, Z ∗ ). A slightlyvariant form of the next result is proved in [8], Theorem 2.17 and Theorem 2.18. Theorem 2.2.
Let X be zero-dimensional and p ∈ β X , then the maximal ideal M pc in C c ( X ) is real if and only if for each f ∈ C ( X, Z ) , f ∗ ( p ) = ω if and only if | M Pc ( f ) | is not infinitely large in C c ( X ) /M pc . Theorem 2.3.
A zero-dimensional space X is c c -realcompact if and only if it is c -realcompact.Proof. Let X be a c -realcompact space and F be a family of closed subsets of X with finite intersection property but with T F = ∅ . To show that X is c c -realcompactwe shall prove that F is not a c -stable family. Indeed { cl β X F : F ∈ F} is a familyof closed subsets of β X with finite intersection property. Since β X is compact,there exists a point p ∈ T F ∈F cl β X F and of course p ∈ β X \ X . Here M pc is afree maximal ideal in C c ( X ). Since X is c -realcompact this implies that M pc is ahyperreal maximal ideal (meaning that it is not a real maximal ideal of C c ( X )).It follows from Theorem 2.2 that there exists f ∈ C ( X, Z ) with f ∗ ( p ) = ω . Since p ∈ cl β X F for each F ∈ F , it is therefore clear that ‘ f ’ is unbounded on each setin the family F . Therefore F is not a c -stable family.Conversely let X be not c -realcompact. Then there exists a real maximal ideal M in C c ( X ), which is not fixed. This means that there is a point p ∈ β X \ X for which M = M pc . Since p ∈ cl β X Z ( f ) for each f ∈ M pc , it follows that { Z ( f ) : f ∈ M pc } is a family of closed sets in X with finite intersection property but withempty intersection. To show that X is not c c -realcompact, it suffices to show that { Z ( f ) : f ∈ M pc } is a c -stable family. So let g ∈ C ( X, Z ). Since M pc is real, thisimplies in view of Theorem 2.2 that g ∗ ( p ) = ω and hence | M pc ( g ) | is not infinitelylarge. It follows therefore from Theorem 2.1 that g is bounded on some Z ( f ) foran f ∈ M pc . This settles that { Z ( f ) : f ∈ M pc } is a c -stable family. (cid:3) By adapting the arguments of Theorem 5.2, Theorem 5.3 and Theorem 5.4 in[9], appropriately we can establish the following facts about c c -realcompact spaceswithout difficulty: Theorem 2.4. (1)
A compact space is c c -realcompact. SUDIP KUMAR ACHARYYA, RAKESH BHARATI, AND A. DEB RAY (2)
A pseudocompact c c -realcompact space is compact. (3) A closed subspace of a c c -realcompact space is c c -realcompact. (4) The product of any set of c c -realcompact spaces is c c -realcompact. (5) If a topological space X = E ∪ F where E is a compact subset of X and F is a Z -embedded c c -realcompact subset of X , meaning that each function in C ( F, Z ) can be extended to a function in C ( X, Z ) , then X is c c -realcompact. (6) A Z -embedded c c -realcompact subset of a Hausdorff space X is a closedsubset of X . We now show that any topological X can be extended to a c c -realcompact spacecontaining the original space X as a C -embedded dense subspace and enjoying adesirable extension property. The proof can be accomplished by closely followingthe arguments adopted to prove Theorem 6.1 in [9]. Nevertheless we give a briefoutline of the main points of proof in our theorem. Theorem 2.5.
Every topological space X can be extended to a c c -realcompact space υ c X as a dense subspace with the following extension property: each continuous mapfrom X into a regular c c -realcompact space Y can be extended to a continuous mapfrom υ c X into Y . X is c c -realcompact if and only if X = υ c X Proof.
For each x ∈ X let G x be the aggregate of all closed sets in X which containthe point x . Then G x is a c -stable family of closed sets in X with finite intersectionproperty and with the prime condition: A ∪ B ∈ G x = ⇒ A ∈ G x or B ∈ G x , A, B ⊆ X . We extend the set X to a bigger set υ c X , so that υ c X \ X becomes anindex set for the collection of all maximal c -stable families of closed subsets of X with finite intersection property but with empty intersection. For each p ∈ υ c X \ X ,let G p designate the corresponding maximal c -stable family of closed sets in X withfinite intersection property and with empty intersection. For each closed set F in X , we write ¯ F = { p ∈ υ c X : F ∈ G p } . Then { ¯ F : F is closed in X } forms a base forclosed sets of some topology on υ c X and in this topology for any closed set F in X ¯ F = cl υ c X F . Since X belongs to each G p , it is clear that X is dense in υ c X . Let t : X → Y be a continuous map with Y , a regular c c -realcompact space. Choose p ∈ υ c X . Let H p = { G ⊆ Y : G is closed in Y and t − ( G ) ∈ G p } . Then H p isa c -stable family of closed sets in Y with finite intersection property. We select apoint y ∈ T H p and we set t ( p ) = y with the aggrement that t ( p ) = t ( p ) in case p ∈ X . Thus t : υ c X → Y is a well defined map which is further continuous. Theremaining parts of the theorem can be proved by making arguments closely as inthe proof of Theorem 6.1 of [9]. (cid:3) c P -compact spaces In this section all the topological spaces X that will appear will be assumed tobe zero-dimensional. We define for any P ∈ Ω( X ), υ P ( X ) = { p ∈ β X : f ∗ ( p ) = ω for each f ∈ C P ( X ) ∩ C ( X, Z ) } . It is clear that if P = E≡ the ideal of all closed setsin X then υ E ( X ) = υ X ≡ { p ∈ β X : f ∗ ( p ) = ω for each f ∈ C ( X, Z ) } the setdefined in the begining of the proof of Theorem 3.8 in [8]. The next theorem putsTheorem 2.3 in a more general setting. Theorem 3.1.
For a
P ∈ Ω , X is c P -compact if and only if X = υ P ( X )We omit the proof of this theorem because it can be done by making someappropriate modification in the arguments adopted in the proof of Theorem 2.3. NTRINSIC CHARACTERIZATIONS OF C -REALCOMPACT SPACES 5 It is clear that if P , Q ∈ Ω( X ) with P ⊂ Q , then any c Q -stable family of closed setsin X is also c P -stable, consequently if X is c P -compact then X is c Q -compact also.In particular every c P -compact space is c c -realcompact and hence c -realcompact inview of Theorem 2.3. The following question therefore seems to be natural. Question 3.2. If X is a zero-dimensional non-compat c -realcompact space, thendoes there exist a minimal ideal P of closed sets in X (minimal in some sense ofthe term) for which X becomes c P -compact ?No possible answer to this question is known to us, however the following proposi-tion shows that the answer to this question is in the negative if the phrase ‘minimal’is interpreted in the set inclusion sense of the term. Theorem 3.3.
Let X be a non compact zero-dimensional space. Suppose P ∈ Ω( X ) is such that X is c P -compact. Then there exists R ∈ Ω( X ) such that R & P and X is c R -compact.Proof. We get from Theorem 3.1 that X = υ P X . As X is non compact we canchoose a point p ∈ β X \ X . Then p / ∈ υ P X . Accordingly there exists f ∈ C P ( X ) ∩ C ( X, Z ) such that f ∗ ( p ) = ω . We select a point x ∈ X such that f ( x ) = 0. Set R = { D ∈ P : x / ∈ D } . It is easy to check that R is an ideal of closed sets in X i.e., R ∈ Ω( X ). Furthermore, cl X ( X − Z ( f )) is a member of P containing the point x .This implies that cl X ( X − Z ( f )) / ∈ R . Thus R P . To show that X is c R -compact.We shall show that X = υ R X (see Theorem 3.1). So choose a point q ∈ β X \ X then q / ∈ υ P X , consequently there exists g ∈ C P ( X ) ∩ C ( X, Z ) such that g ∗ ( q ) = ω .For the distinct points q, x in β X there exist disjoint open sets U , V in this spacesuch that x ∈ U , q ∈ V . Since β X is zero-dimensional there exists therefore aclopen set W in β X such that q ∈ W ⊂ V . The map h : β X → { , } given by h ( W ) = { } and h ( β X \ W ) = { } is continuous. We note that h ( U ) = { } and h ( q ) = 1. Let ψ = h | X . Then ψ ∈ C ( X, Z ). Take l = g.ψ . Since g ∈ C P ( X ) and C P ( X ) is an ideal of C ( X ), it follows that l ∈ C P ( X ). Furthermore the fact that g and ψ are both functions in C ( X, Z ) implies that l ∈ ( X, Z ). Also the function h ∈ C ( β X, Z ) is the unique continuous extension of ψ ∈ C ( X, Z ), hence we canwrite h = ψ ∗ . This implies that l ∗ ( q ) = g ∗ ( q ) ψ ∗ ( q ) = g ∗ ( q ) h ( q ) = ω , because g ∗ ( q ) = ω and h ( q ) = 0On the other hand if y ∈ U ∩ X then h ( y ) = 0 and hence l ( y ) = 0. Since U ∩ X isan open neighbourhood of x in the space X , this implies that x / ∈ cl X ( X \ Z ( l )).Since cl X ( X \ Z ( l )) ∈ P , already verified, it follows that cl X ( X \ Z ( l )) ∈ R . Thus l ∈ C R ( X ) ∩ C ( X, Z ). Since l ∗ ( q ) = ω , this further implies that q / ∈ υ R ( X ). (cid:3) It is trivial that a (zero-dimensional) compact space is c -realcompact. It is alsoobserved that a Lindel¨of space is c -realcompact (Corollary 3.6, [8]). But for aninfinite cardinal number θ , a finally θ -compact space may not be c -realcompact.Indeed the space [0 , ω ) of all countable ordinals is a celebrated example of a zero-dimensional space which is not realcompact (see 8.1, [6]). Since a zero-dimensional c -realcompact space is necessarily realcompact (vide proposition 5.8, [8]) it followstherefore that [0 , ω ) is not a c -realcompact space. But it is easy to show that [0 , ω )is finally ω -compact. For the same reason, the Tychonoff plank T ≡ [0 , ω ) × [0 , ω )- { ( ω , ω ) } of 8.20 in [6], is finally ω -compact without being c -realcompact. It canbe easily shown that a closed subset of a finally θ -compact space is finally θ -compact. SUDIP KUMAR ACHARYYA, RAKESH BHARATI, AND A. DEB RAY
Furthermore, the following characterization of finally θ -compactness of a topo-logical space can be established by routine arguments. Theorem 3.4.
The following two statements are equivalent for an infinite cardinalnumber θ . (1) X is finally θ -compact. (2) If B is a family of closed sets in X , such that for any subfamily B of B with |B | < θ , T B = ∅ , then T B 6 = ∅ . Theorem 3.5.
Let X be c -realcompact and P θ the ideal of all closed finally θ -compact subsets of X . Then X is finally θ -compact if and only if it is c P θ -compact.Proof. Let X be finally θ -compact and p ∈ β X \ X . To show that X is c P θ -compact it suffices to show in view of Theorem 3.1 that p / ∈ υ P θ ( X ). Indeed X is c -realcompact implies that the maximal ideal M pc of C c ( X ) is not real. Consequentlyby Theorem 2.2, there exists f ∈ C ( X, Z ) such that f ∗ ( p ) = ω . Now cl X ( X \ Z ( f )),like any closed subsets of X is finally θ -compact. Thus f ∈ C P θ ( X ) ∩ C ( X, Z ), hence p / ∈ υ P θ ( X ).To prove the converse, let X be not finally θ -compact. It follows from Theorem 3.4that there exists a family B = { B α : α ∈ Λ } of closed sets in X with the followingproperties: for any subfamily B of B with |B | < θ . T B = ∅ but T B = ∅ .Let D = { D α : α ∈ Λ ∗ } be the aggregate of all sets D ′ α s , which are intersectionsof < θ many sets in the family B . Then B ⊆ D and hence T D = ∅ . Also D has finite intersection property. We shall show that D is a c P θ -stable family andhence X is not P θ -compact. Towards such a proof choose f ∈ C P θ ( X ) ∩ C ( X, Z ),then cl X ( X \ Z ( f ) is a finally θ -compact subset of X . Since { X \ B α : α ∈ Λ } is an open cover of X , there exists a subset Λ of Λ with | Λ | < θ such that cl X ( X \ Z ( f )) ⊆ S α ∈ Λ ( X \ B α ). This implies that T α ∈ Λ B α ⊆ Z ( f ) and we notethat T α ∈ Λ B α ∈ D . Thus f becomes bounded on a set lying in the family D . Hence D becomes a c P θ -stable family. (cid:3) NTRINSIC CHARACTERIZATIONS OF C -REALCOMPACT SPACES 7 References [1] S. K. Acharyya and S. K. Ghosh:
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