Ordered field valued continuous functions with countable range
aa r X i v : . [ m a t h . GN ] J u l ORDERED FIELD VALUED CONTINUOUS FUNCTIONSWITH COUNTABLE RANGE
Sudip Kumar Acharyya, Atasi Deb Ray, and Pratip Nandi
Abstract.
For a Hausdorff zero-dimensional topological space X and a totallyordered field F with interval topology, let C c ( X, F ) be the ring of all F − valuedcontinuous functions on X with countable range. It is proved that if F iseither an uncountable field or countable subfield of R , then the structure spaceof C c ( X, F ) is β X , the Banaschewski Compactification of X . The ideals { O p,Fc : p ∈ β X } in C c ( X, F ) are introduced as modified countable analogueof the ideals { O p : p ∈ βX } in C ( X ). It is realized that C c ( X, F ) ∩ C K ( X, F ) = T p ∈ β X \ X O p,Fc , this may be called a countable analogue of the well-knownformula C K ( X ) = T p ∈ βX \ X O p in C ( X ). Furthermore, it is shown that thehypothesis C c ( X, F ) is a Von-Neumann regular ring is equivalent to amongstothers the condition that X is a P − space.
1. Introduction
Let F be a totally ordered field equipped with its ordered topol-ogy. For any topological space X , suppose C ( X, F ) is the set of all F − valued continuous functions on X . This later set becomes a com-mutative lattice ordered ring with unity, if the operations are definedpointwise on X . As in classical scenario with F = R , there is alreadydiscovered an interplay existing between the topological structure of X and the algebraic ring and order structure of C ( X, F ) and a few of itschosen subrings. In order to study this interaction, one can stick toa well-chosen class of spaces viz. the so-called completely F − regulartopological spaces or in brief CF R spaces. X is called CF R spaceif it is Hausdorff and points and closed sets in X could be separatedby F − valued continuous functions in an obvious manner. Problems ofthis kind are addressed in [ ], [ ], [ ], [ ], [ ], [ ]. It turns out thatwith F = R , CF R spaces are precisely zero-dimensional spaces. Thuszero-dimensionality on X can be realized as a kind of separation axiomeffected by F − valued continuous functions on X . In the present arti-cle, we intend to examine the countable analogue of the ring C ( X, F )vis-a-vis the corresponding class of spaces X . Towards that end, we let Mathematics Subject Classification.
Primary 54C40; Secondary 46E25.
Key words and phrases. totally ordered field, zero-dimensional space, Banaschewski Com-pactification, Z cF − ideal, P − space, m cF tpology.The third author thanks the CSIR, New Delhi 110001, India, for financial support. C c ( X, F ) = { f ∈ C ( X, F ) : f ( X ) is a countable subset of F } . Then C c ( X, F ) is a subring as well as a sublattice of C ( X, F ). It is interestingto note that spaces X in which points and closed sets can be separatedby functions in C c ( X, F ) are exactly zero-dimensional also [Theorem2 . C c ( X, F ) endowedwith the well-known Hull-Kernel topology (also known as the structurespace of C c ( X, F )) turns out to be homeomorphic to the BanaschewskiCompactification β X of X [Theorem 2 . F viz. that F is either an uncountable field or a countable subfield of R . A special case of this result choosing F = R reads: the structurespace of the ring C C ( X ) consisting all real-valued continuous functionson X with countable range is β X , which is Remark 3 . ]. Sincethe maximal ideals of C c ( X, F ) can be indexed by virtue of the pointsof β X , it is not surprising that a complete description of these idealscan be given by the family { M p,Fc : p ∈ β X } , where M p,Fc = { f ∈ C c ( X, F ) : p ∈ cl β X Z c ( f ) } , here Z c ( f ) = { x ∈ X : f ( x ) = 0 } standsfor the zero set of f [Remark 2 . . ]. Also, this places Theorem 4 . ] on awider setting. As a natural companion of M p,Fc , we introduce the ideal O p,Fc = { f ∈ M p,Fc : cl β X Z c ( f ) is a neighbourhood of p in β X } .Amongst other facts connecting these two classes of ideals in C c ( X, F ),we have realized that the ideals that lie between O p,Fc and M p,Fc areprecisely those that extend to unique maximal ideals in C c ( X, F ) [The-orem 3 . .
13 in [ ]. Also see Lemma 4 .
11 in [ ] in this connec-tion. If C cK ( X, F ) = { f ∈ C c ( X, F ) : cl X ( X − Z c ( f )) is compact } ,then we have found out a formula for this ring in terms of the ideals O p,Fc as follows : C cK ( X, F ) = T p ∈ β X \ X O p,Fc [in Theorem 3 .
5, comparewith the Theorem 3 .
9, [ ]]. This we may call the appropriate modifiedcountable analogue of the well-known formula in C ( X ) which says that C K ( X ) = T p ∈ βX \ X O p [7 E , [ ]]. The above-mentioned results consti-tute technical section 2 and 3 of the present article.In the final section 4 of this article, we have examined several possibleconsequences of the hypothesis that C c ( X, F ) is a Von-Neumann reg-ular ring with F , either an uncountable field or a countable subfield of R . To aid to this examination, we introduce m Fc − topology on C c ( X, F )as a modified version of m c − topology on C c ( X ) already introduced in[ ]. We establish amongst a host of necessary and sufficient condi-tions that C c ( X, F ) is a Von-Neumann regular ring if and only if eachideal in C c ( X, F ) is closed in the m Fc − topology if and only if X is a P − space. This places theorem 3 . ] on a wider settings, and wemay call it a modified countable analogue of the well-known fact that X is a P − space when and only when each ideal in C ( X ) is closed inthe m − topology [7 Q
4, [ ]]. RDERED FIELD VALUED CONTINUOUS FUNCTIONS WITH COUNTABLE RANGE 3
2. Duality between ideals in C c ( X, F ) and Z F c − filters on X Notation . In spite of the difference of notations, we write for f ∈ C c ( X, F ), Z c ( f ) ≡ { x ∈ X : f ( x ) = 0 } ≡ Z ( f )Let Z c ( X, F ) = { Z c ( f ) : f ∈ C c ( X, F ) } .An ideal unmodified in a ring will always stand for a proper ideal. Definition . A filter of zero sets in the family Z c ( X, F ) is calleda Z F c − filter on X . A Z F c − filter on X is called a Z F c − ultrafilter on X if it is not properly contained in any Z F c − filter on X . Remark . A straight forward use of Zorn’s Lemma tells that a Z F c − filter on X extends to a Z F c − ultrafilter on X . Furthermore anysubfamily of Z c ( X, F ) with finite intersection property can be extendedto a Z F c − ultrafilter on X .The following results correlating Z F c − filters on X and ideals in C c ( X, F ) can be established by using routine arguments.
Theorem . (1) If I is an ideal in C c ( X, F ) , then Z F,C [ I ] = { Z c ( f ) : f ∈ I } isa Z F c − filter on X . Dually for a Z F c − filter F on X , Z − F,C [ F ] = { f ∈ C c ( X, F ) : Z c ( f ) ∈ F } is an ideal in C c ( X, F ) . (2) If M is a maximal ideal in C c ( X, F ) , then Z F,C [ M ] is a Z F c − ultrafilter on X . If U is a Z F c − ultrafilter on X , then Z − F,C [ U ] is a maximal ideal in C c ( X, F ) . Definition . An ideal I in C c ( X, F ) is called Z F c − ideal if Z − F,C [ Z F,C [ I ]] = I It follows from Theorem 2 . C c ( X, F )is a Z F c − ideal. Hence the assignment : M → Z F,C [ M ] establish a one-to-one correspondence between the maximal ideals in C c ( X, F ) and the Z F c − ultrafilters on X .The following propositions can be easily established on using the argu-ments adopted in Chapter 2 and Chapter 4 of [ ] in a straight forwardmanner. Theorem . A Z F c − ideal I in C c ( X, F ) is a prime ideal if andonly if it contains a prime ideal. Hence each prime ideal in C c ( X, F ) ex-tends to a unique maximal ideal, in other words, C c ( X, F ) is a Gelfandring. Theorem . The complete list of fixed maximal ideals in C c ( X, F ) is given by { M cp,F : p ∈ X } where M cp,F = { f ∈ C c ( X, F ) : f ( p ) = 0 } .An ideal I in C c ( X, F ) is called fixed if T f ∈ I Z ( f ) = φ . Definition . X is called countably completely F − regular or inbrief CCF R space if it is Hausdorff and given a closed set K in X anda point x ∈ X \ K , there exists f ∈ C c ( X, F ) such that f ( x ) = 0 and f ( K ) = 1. S.K. ACHARYYA, A. DEB RAY, AND P. NANDI
It is clear that a
CCF R space is
CF R .A CF R space with F = R is zero-dimensional by Theorem 2 . ].A CCF R space with F = R is the same as C − completely regular spaceintroduced in [ ] and is hence zero-dimensional space by Proposition4 . ]. Thus for all choices of the field F , a CCF R space is zero-dimensional. Conversely, it is easy to prove that a zero-dimensionalspace X is CCF R for any totally ordered field F . Thus, the followingresult comes out immediately. Theorem . The statements written below are equivalent for aHausdorff space X and for any totally ordered field F : (1) X is zero-dimensional. (2) X is CCF R . (3) Z c ( X, F ) is a base for closed sets in X . The following result tells that as in the classical situation with F = R , in the study of the ring C c ( X, F ), one can assume without loss ofgenerality that the ambient space X is CCF R , i.e., zero-dimensional.
Theorem . Let X be a topological space and F , a totally or-dered field. Then it is possible to construct a zero-dimensional Haus-dorff space Y such that the ring C c ( X, F ) is isomorphic to the ring C c ( Y, F )We need the following two subsidiary results to prove this theorem.
Lemma . A Hausdorff space X is zero-dimensional if and onlyif given any ordered field F , there exists a subfamily S ⊂ F Xc = { f ∈ F X : f ( X ) is countable set } , which determines the topology on X inthe sense that, the given topology on X is the smallest one with respectto which each function in S is continuous. The proof of this lemma can be accomplished by closely followingthe arguments in Theorem 3 . ] and using Theorem 2 . Lemma . Suppose X is a topological space whose topology isdetermined by a subfamily S of F Xc . Then for a topological space Y ,a function h : Y → X is continuous if and only if for each g ∈ S , g ◦ h : Y → F is a continuous map. The proof of the last lemma is analogous to that of Theorem 3 . ]. Proof. of the main theorem : Define a binary relation ‘ ∼ ’ on X as follows : for x, y ∈ X , x ∼ y if and only if for each f ∈ C c ( X, F ) , f ( x ) = f ( y ).Suppose Y = { [ x ] : x ∈ X } , the set of all corresponding disjointclasses. Let τ : X → Y be the canonical map given by τ ( x ) = [ x ].Each f ∈ C c ( X, F ) gives rise to a function g f : Y → F as follows : RDERED FIELD VALUED CONTINUOUS FUNCTIONS WITH COUNTABLE RANGE 5 g f [ x ] = f ( x ).Let S = { g f : f ∈ C c ( X, F ) } . Then S ⊂ F Yc . Equip Y with the small-est topology, which makes each function in S continuous. It followsfrom the Lemma 2 .
11, that Y is a zero-dimensional space and it is easyto check that Y is Hausdorff. The continuity of τ follows from Lemma2 .
12. Now by the following arguments in Theorem 3 . ], we canprove that the assignment : C c ( Y, F ) → C c ( X, F ) : g → g ◦ τ is anisomorphism onto C c ( X, F ). (cid:3) The following result is a countable counterpart of a portion of The-orem 4 .
11 in [ ]. Theorem . For a zero-dimensional Hausdorff space X and atotally ordered field F , the following three statements are equivalent : (1) X is compact. (2) Each ideal in C c ( X, F ) is fixed. (3) Each maximal ideal in C c ( X, F ) is fixed. Proof. (1) = ⇒ (2) and (2) = ⇒ (3) are trivial. We prove(3) = ⇒ (1) : Let (3) be true.Suppose B is a subfamily of Z c ( X, F ) with finite intersection property.Since Z c ( X, F ) is a base for the closed sets in X (vide Theorem 2 . T B = φ .Indeed B can be extended to a Z F c − ultrafilter U on X . In view ofTheorem 2 .
4, we can write U = Z F,C [ M ] for a maximal ideal M in C c ( X, F ). Hence T B ⊃ T U = φ . (cid:3) Before proceeding further, we reproduce below the following basicfacts about the structure space of a commutative ring with unity from7 M , [ ].Let A be a commutative ring with unity and M ( A ), the set of all max-imal ideals in A . For each a ∈ A , let M a = { M ∈ M ( A ) : a ∈ M } .Then the family {M a : a ∈ A } constitutes a base for the closed sets ofsome topology τ on M ( A ). The topological space ( M ( A ) , τ ) is knownas the structure space of A and is a compact T space. If A is a Gelfandring, then it is established in Theorem 1 .
2, [ ] that τ is a Hausdorfftopology on M ( A ). The closure of a subset M of M ( A ) is givenby : M = { M ∈ M ( A ) : M ⊃ T M } ≡ the hull of the kernel of M . [This is the reason why τ is also called the hull-kernel topologyon M ( A )].Let us denote the structure space of the ring C c ( X, F ) by the notation M c ( X, F ). Since C c ( X, F ) is a Gelfand ring, already verified in Theo-rem 2 .
6, it follows that M c ( X, F ) is a compact Hausdorff space. Fromnow on, we assume that X is Hausdorff and zero-dimensional, and wewill stick to this hypothesis throughout this article. It follows that theassignment ψ : X → M c ( X, F ) given by ψ ( p ) = M cp,F is one-to-one. S.K. ACHARYYA, A. DEB RAY, AND P. NANDI
Furthermore for any f ∈ C c ( X, F ), ψ ( Z c ( f )) = { M cp,F : f ∈ M cp,F } = M f ∩ ψ ( X )where M f = { M ∈ M c ( X, F ) : f ∈ M } .This shows that ψ exchanges the basic closed sets of the two spaces X and ψ ( X ). Finally, ψ ( X ) = { M ∈ M c ( X, F ) : M ⊃ \ ψ ( X ) } = { M ∈ M c ( X, F ) : M ⊃ \ p ∈ X { M cp,F } = { }} = M c ( X, F )This leads to the following proposition :
Theorem . The map ψ : X → M c ( X, F ) given by ψ ( p ) = M cp,F defines a topological embedding of X onto a dense subspace of M c ( X, F ) . In a more formal language, the pair ( ψ, M c ( X, F )) is aHausdorff Compactification of X . The next result shows that the last-mentioned compactification en-joys a special extension property.
Theorem . The compactification ( ψ, M c ( X, F )) enjoys the C − extension property (see Definition . in [ ] ) in the following sense,given a compact Hausdorff zero-dimensional space Y and a continu-ous map f : X → Y , there can be defined a continuous map f c : M c ( X, F ) → Y with the following property : f c ◦ ψ = f . Proof.
This can be accomplished by closely adapting the argu-ments made in the second paragraph in the proof of the Theorem 2 . ]. However, to make the paper self-contained, we sketch a briefoutline of the main points of its proof.Let M ∈ M c ( X, F ). Define as in [ ], f M = { g ∈ C c ( Y, F ) : g ◦ f ∈ M } .Then f M is a prime ideal in C c ( Y, F ). Since C c ( Y, F ) is Gelfand ringand Y is compact and zero-dimensional, it follows from Theorem 2 . y ∈ Y such that T g ∈ f M Z c ( g ) = { y } . Set f c ( M ) = y . Then f c : M c ( X, F ) → Y is the desired continuousmap. (cid:3) Remark . If the structure space M c ( X, F ) of C c ( X, F ) is zero-dimensional, then ( ψ, M c ( X, F )) is topologically equivalent to the Ba-naschewski Compactification β X of X . [see the comments after Defi-nition 2 . ]].We shall now impose a condition on F ; sufficient to make M c ( X, F )zero-dimensional.
Theorem . Suppose the totally ordered field F is either un-countable or a countable subfield of R . Then given f ∈ C c ( X, F ) , there RDERED FIELD VALUED CONTINUOUS FUNCTIONS WITH COUNTABLE RANGE 7 exists an idempotent e ∈ C c ( X, F ) such that e is a multiple of f and (1 − e ) is a multiple of (1 − f ) Proof.
We prove this theorem with the assumption that F is un-countable. The proof for the case when F is a countable subfield of R can be accomplished on using some analogous arguments. We firstassert that the interval [0 ,
1] = { α ∈ F : 0 ≤ α ≤ } is an uncount-able set. This is immediate if F is Archimedean ordered because inthat case F + = { α ∈ F : α ≥ } = S n ∈ N ∪{ } [ n, n + 1] and for each n ∈ N ∪ { } , [ n, n + 1] is equipotent with [0 ,
1] through the translationmap : α → ( α + n ) , α ∈ [0 , F is non-Archimedeanordered field. If possible let [0 ,
1] be a countable set. Then the set F + \ S n ∈ N ∪{ } [ n, n + 1] becomes an uncountable set, which means thatthe set of all infinitely large members of F make an uncountable set.Consequently, the set I = { α ∈ F + : 0 < α < n f or each n ∈ N } comprising of the infinitely small members of F is an uncountable set.But it is easy to see that I ⊂ (0 ,
1) and therefore (0 ,
1) turns out tobe an uncountable set – a contradiction. Thus it is proved that [0 , α > F , (0 , α )becomes an uncountable set]. So we can choose r ∈ (0 ,
1) such that r / ∈ f ( X ). Let, W = { x ∈ X : f ( x ) < r } = { x ∈ X : f ( x ) ≤ r } andso X \ W = { x ∈ X : f ( x ) > r } = { x ∈ X : f ( x ) ≥ r } . It is clear that W and X \ W are clopen sets in X and the function e : X → F definedby e ( W ) = { } and e ( X \ W ) = { } is an idempotent in C c ( X, F ). Wesee that Z c ( f ) ⊂ Z c ( e ) and Z c (1 − f ) ⊂ Z c (1 − e ) and we can say that Z c ( e ) is a neighbourhood of Z c ( f ) and Z c (1 − e ) is a neighbourhood of Z c (1 − f ) in the space X . Hence e is a multiple of f and (1 − e ) is amultiple of (1 − f ). [compare with the arguments made in Remark 3 . ]]. (cid:3) Theorem . The structure space M c ( X, F ) of C c ( X, F ) is zero-dimensional and hence M c ( X, F ) = β X . [Here F is either uncountable or a countable subfield of R ] Proof.
Recall the notation for f ∈ C c ( X, F ), M f = { M ∈ M c ( X,F ) : f ∈ M } . Suppose M ∈ M c ( X, F ) is such that
M / ∈ M f . Itsuffices to find out an idempotent e in C c ( X, F ) with the property : M f ⊂ M e and M / ∈ M e . The simple reason is that e. (1 − e ) = e − e = e − e = 0 and hence M e = M c ( X, F ) \M (1 − e ) , consequently M e is aclopen set in M c ( X, F ). Now towards finding out such an idempotentlet us observe that
M / ∈ M f implies that f / ∈ M , which further impliesthat < f, M > = C c ( X, F ). Hence we can write : 1 = f.h + g , where h ∈ C c ( X, F ) and g ∈ M . By Theorem 2 .
17, there exists an idempo-tent e in C c ( X, F ) such that e = g .g and (1 − e ) = g . (1 − g ), where g , g ∈ C c ( X, F ). Now let N ∈ M f , then f ∈ N and so f.h ∈ N ,which implies that (1 − g ) ∈ N consequently (1 − e ) ∈ N . Therefore S.K. ACHARYYA, A. DEB RAY, AND P. NANDI e / ∈ N , which means that N / ∈ M e , i.e., N ∈ M c ( X, F ) \M e . Againsince g ∈ M , it follows that e ∈ M , thus M ∈ M e . (cid:3) Remark . On choosing F = R and X = Q in the above The-orem 2 .
18, we get that β Q = structure space of C ( Q , R ) = β Q . Thus β Q becomes zero-dimensional, i.e., Q is strongly zero-dimensional.This is a standard result in General Topology – indeed a Lindel¨of zero-dimensional space is strongly zero-dimensional. [Theorem 6 . .
7, [ ]].One of the major achievements in the theory of C ( X ) is that acomplete description of the maximal ideals in this ring can be given.This is a remark made in the beginning of Chapter 6 in [ ]. In orderto give such a description, it becomes convenient to archive βX as thespace of Z − ultrafilter on X equipped with the Stone- topology andformal construction of such a thing is dealt in rigorously in Chapter 6in [ ]. We follow the same technique in order to furnish an explicitdescription of maximal ideals in C c ( X, F ).For each p ∈ X , let A cp,F = { Z ∈ Z c ( X, F ) : p ∈ Z } ≡ Z F,C [ M cp,F ].Thus X is a readymade index set for the family of fixed Z F c − ultrafilterson X . As in Chapter 6, [ ], we extend the set X to a set αX to serveas an index set for the family of all Z F c − ultrafilters on X . For p ∈ αX ,let the corresponding Z F c − ultrafilter be designated as A p,Fc with theunderstanding that if p ∈ X , then A p,Fc = A cp,F .For Z ∈ Z c ( X, F ), let Z = { p ∈ αX : Z ∈ A p,Fc } . Then { Z : z ∈ Z c ( X, F ) } makes a base for the closed sets of some topology on αX in which for Z ∈ Z c ( X, F ), Z = cl αX Z . Furthermore, for Z , Z ∈ Z c ( X, F ), Z ∩ Z = Z ∩ Z and αX becomes a compact Hausdorffspace containing X as a dense subset. Also given a point p ∈ αX , A p,Fc is the unique Z F c − ultrafilter on X which converges to p and finally αX possesses the C − extension property meaning that if Y is a compactHausdorff zero-dimensional space and f : X → Y , a continuous map,then f can be extended to a continuous map f : αX → Y . All thesefacts can be realized just by closely following the arguments in Chapter6 in [ ]. Theorem . The space αX is a zero-dimensional space. [Blanket assumption: F is either an uncountable field or a countablesubfield of R ] Proof.
Let p ∈ αX and Z ∈ Z c ( X, F ) be such that p / ∈ Z .It suffices to find out a clopen set K in αX such that Z ⊂ K and p / ∈ K .Now p / ∈ Z = ⇒ Z / ∈ A p,Fc . Since A p,Fc is a Z F c − ultrafilter, this impliesthat there exists Z ∗ ∈ A p,Fc such that Z ∩ Z ∗ = φ . Hence there exists f ∈ C c ( X, F ) such that f : X → [0 ,
1] in F such that f ( Z ∗ ) = { } and f ( Z ) = { } . Using the hypothesis that F is an uncountable field, andtake note of the arguments in the proof of the Theorem 2 .
17, we can
RDERED FIELD VALUED CONTINUOUS FUNCTIONS WITH COUNTABLE RANGE 9 find out an r ∈ (0 ,
1) in F such that r / ∈ f ( X ) [analogous argumentscan be made if F is a countable subfield of R ].Let K = { x ∈ X : f ( x ) > r } = { x ∈ X : f ( x ) ≥ r } . Then K is aclopen set in X containing Z and therefore Z ⊂ K . Now Z ∗ ⊂ X \ K implies Z ∗ ∩ K = φ and hence Z ∗ ∩ K = φ , i.e., Z ∗ ∩ K = φ . Since Z ∗ ∈ A p,Fc and therefore p ∈ Z ∗ , this further implies that p / ∈ K . (cid:3) Remark . Since αX enjoys the C − extension property andis zero-dimensional, it follows from Definition 2 . ] that αX isessentially the same as β X , the Banaschewski Compactification of X and hence we can write for any p ∈ β X and Z ∈ Z c ( X, F ), Z ∈ A p,Fc if and only if p ∈ cl β X . If we now write M p,Fc = Z − F,C [ A p,Fc ], then thisbecomes a maximal ideal in C c ( X, F ). Since by Theorem 2 . C c ( X, F ) and Z F c − ultrafilters on X via the map M → Z F,C [ M ], acomplete description of the maximal ideals in C c ( X, F ) is given by thelist { M p,Fc : p ∈ β X } where M p,Fc = { f ∈ c c ( X, F ) : p ∈ cl β X Z c ( f ) }
3. The ideals O p,Fc and a formula for C cK ( X, F )For each p ∈ β X , set O p,Fc = { f ∈ C c ( X, F ) : cl β X Z c ( f ) is a neighbourhood of p in β X } Then the following facts come out as modified countable analogue ofthe relations between the ideals M p and O p in the classical scenariorecorded in 7 .
12, 7 .
13, 7 .
15 in [ ]. Also see Lemma 4 .
11 in [ ] in thisconnection. Theorem . Let the ordered field F be either uncountable or acountable subfield of R . Then for a zero-dimensional Hausdorff space X , the following statements are true : (1) O p,Fc is a Z F c − ideal in C c ( X, F ) contained in M p,Fc . (2) O p,Fc = { f ∈ C c ( X, F ) : there exists an open neighbourhood Vof p in β X such that Z c ( f ) ⊃ V ∩ X } . (3) For p ∈ β X and f ∈ C c ( X, F ) , f ∈ O p,Fc if and only if thereexists g ∈ C c ( X, F ) \ M p,Fc such that f.g = 0 , hence each non-zero element in O p,Fc is a divisor of zero in C c ( X, F ) . Indeed O p,Fc is a z o − ideal in C c ( X, F ) . (4) An ideal I in C c ( X, F ) is extendable to a unique maximal idealif and only if there exists p ∈ β X such that O p,Fc ⊂ I . (5) For p ∈ β X , O p,Fc is a fixed ideal if and only if p ∈ X . Proof.
The statements (1), (2) and (4) can be proved by makingarguments parallel to those adopted to prove the corresponding resultsin the classical situation with F = R in Sections 7 . , . , .
15 in [ ].We prove only the statements (3) and (5).To prove (3), let f ∈ O p,Fc . Then by (2), there exists an open neigh-bourhood V of p in β X such that Z c ( f ) ⊃ V ∩ X . Since β X is zero dimensional, there exists a clopen set K in β X such that β X \ V ⊂ K and p / ∈ K . The function h : β X → F , defined by h ( K ) = { } and h ( β X \ K ) = { } belongs to C c ( β X, F ). Take g = h | X . Then g ∈ C c ( X, F ), f.g = 0 and p / ∈ cl β X Z c ( g ), hence g / ∈ M p,Fc .Conversely let there exist g ∈ C c ( X, F ) \ M p,Fc such that f.g = 0. Then p / ∈ cl β X Z c ( g ). Therefore there exists an open neighbourhood V of p in β X such that V ∩ Z c ( g ) = φ . Since Z c ( f ) ∪ Z c ( g ) = X , it followsthat X ∩ V ⊂ Z c ( f ). Hence from (2), we get that f ∈ O p,Fc .To prove the last part of (3), we recall that an ideal I in a commuta-tive ring A with unity is called a z o − ideal if for each a ∈ I , P a ⊂ I ,where P a is the intersection of all minimal prime ideals in A contain-ing a . We reproduce the following useful formula from Proposition 1 . ], which is also recorded in Theorem 3 .
10 in [ ] : if A is a re-duced ring meaning that 0 is the only nilpotent member of A , then P a = { b ∈ A : Ann ( a ) ⊂ Ann ( b ) } , where Ann ( a ) = { c ∈ A : a.c = 0 } is the annihilator of a in A . Hence for any f ∈ C c ( X, F ), P f ≡ the intersection of all minimal prime ideals in C c ( X, F ) which con-tain f = { g ∈ C c ( X, F ) :
Ann ( f ) ⊂ Ann ( g ) } .Now to show that O p,Fc is a z o − ideal in C c ( X, F ), for any p ∈ β X ,choose f ∈ O p,Fc and g ∈ P f . Therefore Ann ( f ) ⊂ Ann ( g ). Butfrom the result (3), we see that there exists h ∈ C c ( X, F ) \ M p,Fc suchthat f.h = 0 and hence h ∈ Ann ( f ). Consequently, h ∈ Ann ( g ), i.e., g.h = 0. Thus P f ⊂ O p,Fc and hence O p,Fc is a z o − ideal in C c ( X, F ).Proof of (5) : If p ∈ X , then M p,Fc = M cp,F , a fixed ideal, hence O p,Fc isalso fixed.Now let p ∈ β X \ X . Choose x ∈ X and a closed neighbourhood W of p in β X such that x / ∈ W . Since β X is zero-dimensional, thereexists a clopen set K in β X such that W ⊂ K and x / ∈ K . Let g : β X → F be defined by g ( K ) = { } and g ( β X \ K ) = { } . Then g ∈ C c ( β X, F ) and hence h = g | X ∈ C c ( X, F ). We observe that h ( x ) = 1 and Z c ( h ) ⊃ K ∩ X . It follows from the result (2) that h ∈ O p,Fc . This proves that O p,Fc is a free ideal in C c ( X, F ) (cid:3) The following properties of C cK ( X, F ) = { f ∈ C c ( X, F ) : f hascompact support i.e. cl X ( X \ Z c ( f )) is compact } can be established asparallel to the analogous properties of the ring C K ( X ) = { f ∈ C ( X ) : f has compact support } given in 4 D , [ ]. Theorem . Let X be Hausdorff and zero-dimensional. Then : (1) C cK ( X, F ) ⊂ C c ( X, F ) ∩ C ∗ ( X, F ) , where C ∗ ( X, F ) = { f ∈ C ( X, F ) : cl F f ( X ) is compact } and equality holds if and onlyif X is compact. (2) If X is non-compact, then C cK ( X, F ) is an ideal(proper) of C c ( X, F ) . RDERED FIELD VALUED CONTINUOUS FUNCTIONS WITH COUNTABLE RANGE 11 (3) C cK ( X, F ) is contained in every free ideal of C c ( X, F ) . C cK ( X, F ) itself is a free ideal of C c ( X, F ) if and only if X is non-compactand locally compact. (4) X is nowhere locally compact if and only if C cK ( X, F ) = { } and this is the case when and only when β X \ X is dense in β X . [Compare with F , [ ] ] Remark . C cK ( X, F ) ⊂ T { O p,Fc : p ∈ β X \ X } .This follows from Theorem 3 . . Theorem . Let f ∈ C c ( X, F ) be such that cl β X Z c ( f ) is a neigh-bourhood of β X \ X . Then f ∈ C cK ( X, F ) . Proof.
It suffices to show that supp ( f ) ≡ cl X ( X \ Z c ( f )) is closedin β X and hence compact. As Z c ( f ) is closed in X , it follows that cl β X Z c ( f ) ∩ ( X \ Z c ( f )) = φ . The hypothesis tells that there existsan open set W in β X such that β X \ X ⊂ W ⊂ cl β X Z c ( f ). Hence W ∩ ( X \ Z c ( f )) = φ , which further implies because W is open in β X that W ∩ cl β X ( X \ Z c ( f )) = φ . Consequently W ∩ cl X ( X \ Z c ( f )) = φ .Since β X \ X ⊂ W , it follows therefore that no point of β X \ X is alimit point of cl X ( X \ Z c ( f )) in the space β X . Thus there does notexist any limiting point of cl X ( X \ Z c ( f )) outside it in the entire space β X . Hence cl X ( X \ Z c ( f )) is closed in β X . (cid:3) Theorem . Let X be zero-dimension and Hausdorff. Then C cK ( X, F ) = T { O p,Fc : p ∈ β X \ X } Proof.
Let f ∈ O p,Fc for each p ∈ β X \ X . Then cl β X Z c ( f )is a neighbourhood of each point of β X \ X in the space β X . Itfollows from Theorem 3 . f ∈ C cK ( X, F ). Thus T { O p,Fc : p ∈ β X \ X } ⊂ C cK ( X, F ). The reversed implication relation is alreadyrealized in Remark 3 .
3. Hence C cK ( X, F ) = T { O p,Fc : p ∈ β X \ X } . (cid:3)
4. Von Neumann regularity of C c ( X, F ) versus P − space X We recall from [ ] that X is called P F − space if C ( X, F ) is a Von-Neumann regular ring. By borrowing the terminology from [ ], wecall a zero-dimensional space X , a countably P F − space or CP F − spaceif C c ( X, F ) is Von-Neumann regular ring. Thus in this terminology, CP R − spaces are precisely CP − spaces introduced in [ ], Definition5 .
1. It is still undecided whether there exist an ordered field F and azero-dimensional space X for which X is a P F − space without being a P − space (see the comments preceding Definition 3 . ]). However,we shall prove that subject to the restrictions imposed on the field F ,already used several times in this paper, CP F − spaces and P − spacesare one and the same. We want to mention in this context that the zero set of a function f in C ( X, F ) may not be a G δ − set [see Theorem2 . ]]. In contrast, we shall show that the zero set of a functionlying in C c ( X, F ) is necessarily a G δ − set. Before proceeding further,we make the assumption throughout the rest of this article that theordered field F is either uncountable or a countable subfield of R . Theorem . A zero set Z ∈ Z c ( X, F ) is a G δ − set. Proof.
We can write Z = Z c ( f ) for some f ≥ C c ( X, F ).Since f ( X ) is a countable subset of F , we can write, f ( X ) \{ } = { r , r , ..., r n , ... } ; a countable set in F + . It follows that Z c ( f ) = T ∞ n =1 f − ( − r n , r n ) = a G δ − set in X . (cid:3) The following results are generalized versions of Proposition 4 . . . ]. Theorem . If A and B are disjoint closed sets in X with A ,compact, then there exists f ∈ C c ( X, F ) such that f ( A ) = { } and f ( B ) = { } Theorem . For f ∈ C c ( X, F ) , Z c ( f ) is a countable intersectionof clopen sets in X Proof.
As in the proof of Theorem 4 .
1, we can assume f ≥ f ( X ) \{ } = { r , r , ..., r n , ... } . Now if F is an uncountable orderedfield, then for each n ∈ N , we can choose s n ∈ F such that 0 < s n < r n and s n / ∈ { r , r , ..., r n , ... } . On the other hand, if F is a countablesubfield of R , then we can pick up for each n ∈ N an irrational pointdenoted by the same symbol s n with the above-mentioned condition,i.e., 0 < s n < r n and s n / ∈ { r , r , ..., r n , ... } . It follows that Z c ( f ) = T ∞ n =1 f − ( − s n , s n ) = T ∞ n =1 f − [ − s n , s n ] = a countable intersection ofclopen sets in X . (cid:3) Theorem . A countable intersection of clopen sets in X is azero set in Z c ( X, F ) (equivalently, a countable union of clopen sets in X is a co-zero set, i.e., the complement in X of a zero set in Z c ( X, F ) ). Proof.
Since in any topological space, a countable union of clopensets can be expressed as a countable union of pairwise disjoint clopensets, we can start with a countable family { L i } ∞ i =1 of pairwise disjointclopen sets in X . For each n ∈ N , define a function e n : X → F asfollows : e n ( L n ) = { } and e n ( X \ L n ) = { } . Then e n ∈ C c ( X, F )and is an idempotent in this ring. Furthermore, it is easy to see thatif m = n , then e m .e n = 0. Let h ( x ) = P ∞ n =1 e n ( x )3 n , x ∈ X . Then h : X → F is a continuous function and h ( X ) ⊂ { , , , ... } . Thus h ∈ C c ( X, F ). It is clear that S ∞ n =1 L n = X \ Z c ( h ). (cid:3) Theorem . Z c ( X, F ) is closed under countable intersection. Proof.
Follows from Theorem 4 . . (cid:3) RDERED FIELD VALUED CONTINUOUS FUNCTIONS WITH COUNTABLE RANGE 13
Theorem . Suppose a compact set K in X is contained in a G δ − set G . Then there exists a zero set Z in Z c ( X, F ) such that K ⊂ Z ⊂ G . Proof.
We can write G = T ∞ n =1 W n where each W n is open in X .For each n ∈ N , K and X \ W n are disjoint closed sets in X with K compact. Hence by Theorem 4 .
2, there exists g n ∈ C c ( X, F ) such that g n ( K ) = { } and g n ( X \ W n ) = { } . It follows that K ⊂ Z c ( g n ) ⊂ W n for each n ∈ N . Consequently, K ⊂ T ∞ n =1 Z c ( g n ) ⊂ G . But by Theorem4 .
5, we can write T ∞ n =1 Z c ( g n ) = Z c ( g ) for some g ∈ C c ( X, F ). Hence K ⊂ Z c ( g ) ⊂ G . (cid:3) Before giving several equivalent descriptions of the defining prop-erty of CP F − space in the manner 4 J of [ ] and the theorem 5 . ]. We like to introduce a suitable modified countable version of m − topology on C ( X ) as dealt with in 2 N , [ ].For each g ∈ C c ( X, F ) and a positive unit u in this ring, set M F ( g, u ) = { f ∈ C c ( X, F ) : | f ( x ) − g ( x ) | < u ( x ) f or each x ∈ X } . Then it can beproved by routine computation that { M F ( g, u ) : g ∈ C c ( X, F ) , u, apositive unit in C c ( X, F ) } is an open set for some topology on C c ( X, F ),which we call m Fc − topology on C c ( X, F ). A special case of this topol-ogy with F = R is already considered in [ ], Section 3. The followingtwo can be established by making straight forward modifications in thearguments adopted to prove Theorem 3 . . ]. Theorem . Each maximal ideal in C c ( X, F ) is closed in the m Fc -topology. Theorem . For any ideal I in C c ( X, F ) , its closure in m Fc − topology is given by : I = T { M p,Fc : p ∈ β X and M p,Fc ⊃ I } ≡ theintersection of all the maximal ideals in C c ( X, F ) which contains I . [compare with 7 Q
2, [ ]] Theorem . An ideal I in C c ( X, F ) is closed in m Fc − topology ifand only if it is the intersection of all the maximal ideals in this ringwhich contains I . [This follows immediately from Theorem 4 . X is a CP F − space. Theorem . Let X be a zero-dimensional Hausdorff space and F , a totally ordered field with the property mentioned in the beginningof this section. Then the following statements are equivalent : (1) X is a CP F − space. (2) Each zero set in Z c ( X, F ) is open. (3) Each ideal in C c ( X, F ) is a Z F c − ideal. (4) For all f, g in C c ( X, F ) , < f, g > = < f + g > . (5) Each prime ideal in C c ( X, F ) is maximal. (6) For each p ∈ X , M cp,F = O cp,F . (7) For each p ∈ β X , M p,Fc = O p,Fc . (8) Each ideal in C c ( X, F ) is the intersection of all the maximalideals containing it. (9) Each G δ − set in X is open (which eventually tells that X is a P − space) (10) Every ideal in C c ( X, F ) is closed in the m Fc − topology. Proof.
Equivalence of the first eight statements can be proved bymaking an almost repetition of the arguments to prove the equivalenceof the analogous statements in 4 J , [ ] [Also see the Theorem 5 . ]]. We prove the equivalence of the statements (2) , (9) , (10).(9) = ⇒ (2) is immediate because of Theorem 4 . ⇒ (9) : Let (2) be true and G be a non-empty G δ − set in X .Then by Theorem 4 .
6, for each point x ∈ G , there exists a zero set Z x ∈ Z c ( X, F ) such that x ∈ Z x ⊂ G . Since Z x is open in X by (2),it follows that x is an interior point of G . In other words, G is open in X .Equivalence of (8) and (10) follows from Theorem 4 . (cid:3) Remark . On choosing F = Q in Theorem 4 .
10, we get that azero dimensional space X is a P − space if and only if C ( X, Q ) is a VonNeumann regular ring, i.e., X is a P Q − space. Thus each P Q − spaceis a P − space. But we note that, though the cofinality character of Q is ω , it is not Cauchy complete. This improves the conclusion of theTheorem 3 . ], which says that if F is a Cauchy complete totallyordered field with cofinality character ω , then every P F − space is a P − space.Open question : If p ∈ β X , then does the set of prime ideals in C c ( X, F ) that lie between O p,Fc and M p,Fc make a chain? References [1] S.K. Acharyya, K.C. Chattopadhyaya and P.P. Ghosh :
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Department of Pure Mathematics, University of Calcutta, 35, Ballygunge Circu-lar Road, Kolkata 700019, West Bengal, India
E-mail address : [email protected] Department of Pure Mathematics, University of Calcutta, 35, Ballygunge Circu-lar Road, Kolkata 700019, West Bengal, India
E-mail address : [email protected] Department of Pure Mathematics, University of Calcutta, 35, Ballygunge Circu-lar Road, Kolkata 700019, West Bengal, India
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