Some generalizations and unifications of C K (X) , C ψ (X) and C ∞ (X)
aa r X i v : . [ m a t h . GN ] J a n Some generalizations and unifications of C K ( X ), C ψ ( X ) and C ∞ ( X ) A. TaherifarDepartment of Mathematics, Yasouj University, Yasouj , [email protected]
Dedicated to Professor Azarpanah
Abstract.
Let P be an open filter base for a filter F on X . We denote by C P ( X ) ( C ∞P ( X )) the set of all functions f ∈ C ( X ) where Z ( f ) ( { x : | f ( x ) | < n } ) contains an element of P . First, we observe that every proper subringsin the sense of Acharyya and Ghosh (Topology Proc. 2010) has such formand vice versa. After wards, we generalize some well known theorems about C K ( X ) , C ψ ( X ) and C ∞ ( X ) for C P ( X ) and C ∞P ( X ). We observe that C ∞P ( X )may not be an ideal of C ( X ). It is shown that C ∞P ( X ) is an ideal of C ( X )and for each F ∈ F , X \ F is bounded if and only if the set of non-clusterpoints of the filter F is bounded. By this result, we investigate topologicalspaces for which C ∞P ( X ) is an ideal of C ( X ) whenever P = { A ( X : A isopen and X \ A is bounded } (resp., P = { A ( X : X \ A is finite } ). Moreover,we prove that C P ( X ) is an essential (resp., free) ideal if and only if the set { V : V is open and X \ V ∈ F } is a π -base for X (resp., F has no clusterpoint). Finally, the filter F for which C ∞P ( X ) is a regular ring (resp., z -ideal)is characterized. Keywords: local space, bounded subset, z -ideal, regular ring, essential ideal, F - CG δ subset. C . In this paper, X assumed to be a completely regular Hausdorff space. C ( X )( C ∗ ( X ))stands for the ring of all real valued (bounded) continuous functions on X . Asubcollection B of a filter F is a filter base for F if and only if each element of1 ome generalizations and unifications of C K ( X ) , C ψ ( X ) and C ∞ ( X ) 2 F contains some element of B . A nonempty collection B of nonempty subsetsof the space X is a filter base for some filter on X if and only if the intersectionof any two elements of B contains an element of B . If any element of B is anopen subset, we say B is an open filter base. We shall say that a point p ∈ X is a cluster point of F if every neighborhood of p meets every member of F , inother words, p ∈ T F ∈F F . Kohls proved in [13] that the intersection of all freemaximal ideals in C ∗ ( X ) is precisely the set C ∞ ( X ) consists of all continuousfunctions f ∈ C ( X ) which vanish at infinity, in the sense that { x : | f ( x ) | ≥ n } is compact for each n ∈ N . Kohls also shown in [13] that the C K ( X ) consistsof all functions in C ( X ) with compact support is the intersection of all freeideals in C ( X ) and all free ideals in C ∗ ( X ). It is well known that C K ( X ) isan ideal of C ( X ) and it is easy to see that C ∞ ( X ) is an ideal of C ∗ ( X ) butnot in C ( X ), see [6]. An ideal I of C ( X ) is called essential ideal if I ∩ ( f ) = 0,for every non-zero element f ∈ C ( X ). An ideal I of C ( X ) is called z -ideal if Z ( f ) ⊆ Z ( g ), f ∈ I , then g ∈ I . In particular, maximal ideals, minimal primeideals, and most of familiar ideals in C ( X ) are z -ideals, see [3] and [7].Our aim of this paper is to reveal some important properties of a specialkind of generalized form of C K ( X ) and C ∞ ( X )), which is denoted by C P ( X )and C ∞P ( X ). In section 2, some examples of these subrings are given andwe prove that for any open filter base P , there is an ideal P ′ of closed setssuch that C P ( X ) = C P ′ ( X ) and C ∞P ( X ) = C P ′ ∞ ( X ) and vice versa whenever C P ′ ∞ ( X ) is a proper subsring pf C ( X ) (see [1]). It is shown that C P ( X ) is afree ideal if and only if F has no cluster point. Consequently, we observe that X is a local space (i,e., there is an open filter base P for a filter F which F has no cluster point) if and only if there is an open filter base P such that C P ( X ) is a free ideal. In this section, we show that C P ( X ) (resp., C ∞P ( X ))is a zero ideal if and only if any element of P ( F ) is dense in X . A subset A of X is called an F - CG δ , if A = T ∞ i =1 A i , where each A i is an open subset, X \ A i and A i +1 are completely separated and each A i ∈ F . We prove thatthat C P ( X ) = C ∞P ( X ) if and only if every closed F - CG δ is an element of F .We give an example of an open filter base P for a filter F such that C ∞P ( X )is not an ideal of C ( X ). It is also shown that C ∞P ( X ) is an ideal of C ( X )and for any F ∈ F , X \ F is bounded if and only if the set of non-clusterpoints of the filter F is bounded which is a generalization of Theorem 1.3 in[6]. Consequently, if X is a pseudocompact space, then for any open filter base P , C ∞P ( X ) is an ideal of C ( X ).Section 3 is devoted to the essentiality of C P ( X ) and ideals in C ∞P ( X ).In [5], it was proved that an ideal I of C ( X ) is an essential ideal if and onlyif T Z [ I ] does not contain an open subset (i.e., int T Z [ I ] = ∅ ). Azarpanahin [4], proved that C K ( X ) is an essential ideal if and only if X is an almostlocally compact (i.e, every non-empty open set of X contains a non-emptyopen set with compact closure). We generalize these results for C P ( X ) and ome generalizations and unifications of C K ( X ) , C ψ ( X ) and C ∞ ( X ) 3 C ∞P ( X ). It is proved that an ideal E in C ∞P ( X ) is an essential ideal if andonly if T Z [ E ] does not contain a subset V where X \ V ∈ F and intV = ∅ .We also prove that C P ( X ) is an essential ideal if and only if the set { V : V is open and X \ V ∈ F } is a π -base for X .In section 4, it is shown that C ∞P ( X ) is a z -ideal if and only if everycozero-set containing a closed F - CG δ is an element of F . Also we show that C ∞P ( X ) is a regular ring (in the sense of Von Neumann) if and only if everyclosed F - CG δ is an open subset and belongs to F . Finally, we see that if P = { A : A is open and X \ A is Lindel¨of } and X is a non-Lindel¨of space, then P is an open filter and C ∞P ( X ) is a regular ring if and only if every closed P - CG δ is an open subset. C P ( X ) and C ∞P ( X ) Let P be an open filter base for a filter F on topological space X , we denote by C P ( X ) the set of all functions f in C ( X ) for which Z ( f ) contains an elementof P . Also C ∞P ( X ) denotes the family of all functions f ∈ C ( X ) for whichthe set { x : | f ( x ) | < n } contains an element of P , for each n ∈ N .Recall that, for a subset A of X , O A = { f : A ⊆ intZ ( f ) } . Lemma 2.1.
The following statements hold.1. C P ( X ) is a z -ideal of C ( X ) contained in C ∞P ( X ).2. C P ( X ) = P A ∈P O A = S A ∈P O A C ∞P ( X ) is a proper subring of C ( X ). Proof. (1) By definition of C P ( X ) and since P is a base filter, it is easy tosee that C P ( X ) is a z -ideal and contained in C ∞P ( X ).(2) Let f ∈ C P ( X ). Then there exists A ∈ P such that A ⊆ Z ( f ).Hence A ⊆ intZ ( f ), i.e., f ∈ O A ⊆ P A ∈P O A . If f ∈ P A ∈P O A , then thereare 1 ≤ i ≤ n and f i ∈ O A i such that f = f + ... + f n , thus T ni =1 A i ⊆ T ni =1 intZ ( f i ) ⊆ Z ( f ). But T ni =1 A i contains an element of P , so f ∈ C P ( X ).The proof of the second equality is obvious.(3) First, we observe that C ∞P ( X ) is a proper subset of C ( X ). For, if C ∞P ( X ) = C ( X ), then ∅ ∈ P , which is a contradiction. On the other hand,we have { x : | f ( x ) − g ( x ) | < n } ⊇ { x : | f ( x ) | < n } ∩ { x : | g ( x ) | < n } and { x : | f ( x ) g ( x ) | < n } ⊇ { x : | f ( x ) | < √ n } ∩ { x : | g ( x ) | < √ n } . ome generalizations and unifications of C K ( X ) , C ψ ( X ) and C ∞ ( X ) 4Therefore, C ∞P ( X ) is a proper subring of C ( X ). (cid:3) Recall that a family P of closed subsets of X is called an ideal of closedsets in X , if it satisfies in the following conditions.1. If A, B ∈ P , then A ∪ B ∈ P .2. If A ∈ P and B ⊆ A with B closed in X , then B ∈ P .In [1], Acharyya and Ghosh for ideal P of closed subsets of X defined C P ( X )and C P∞ ( X ) as follows; C P ( X ) = { f ∈ C ( X ) : cl ( X \ Z ( f )) ∈ P} ; and C P∞ ( X ) = { f ∈ C ( X ) : { x : | f ( x ) | ≥ n } ∈ P , for each, n ∈ N } . In the next result we give a new presentation of these subrings. We notethat for ideal P of closed sets, C P∞ ( X ) may be C ( X ) but by Proposition 2.1, C ∞P ( X ) for each open filter P is a proper subring. Proposition 2.2.
The following statements hold.1. For every open filter base P , there exists an ideal P ′ of closed sets suchthat C P ( X ) = C P ′ ( X ) and C ∞P ( X ) = C P ′ ∞ ( X ).2. If C P∞ ( X ) is a proper subring of C ( X ), then there is an open filter base Q such that C P ( X ) = C Q ( X ) and C P∞ ( X ) = C ∞Q ( X ). Proof. (1) Let P be an open filter base. Consider P ′ as follows; P ′ = { A : A is closed and A ⊆ X \ B for some B ∈ P} .Then, it is easy to see that P ′ is an ideal of closed sets in X , C P ( X ) = C P ′ ( X )and C ∞P ( X ) = C P ′ ∞ ( X ).(2) Assume that C P∞ ( X ) is a proper subring of C ( X ) and Q = { A ⊆ X:X \ A ∈ P} . Then we can see that Q is an open filter, C P ( X ) = C Q ( X ) and C P∞ ( X ) = C ∞Q ( X ). (cid:3) Example 2.3.
Let X be a non-compact Hausdorff space and P = { A ( X : X \ A is compact } . Then P is an open filter base , C P ( X ) = C K ( X ) and C ∞P ( X ) = C ∞ ( X ). Example 2.4.
Let P = { A : A is open and X \ A is Lindel¨of } and X be anon-Lindel¨of space. Then P is an open filter and we have C ∞P ( X )= { f : X \ Z ( f ) is a Lindel¨of subset of X } and C P ( X )= { f : X \ Z ( f ) is a Lindel¨of subset of X } . ome generalizations and unifications of C K ( X ) , C ψ ( X ) and C ∞ ( X ) 5To see this, let f ∈ C ∞P ( X ). Then for each n ∈ N , { x : | f ( x ) | ≥ n } ⊆ X \ A for some A ∈ P , so it is a Lindel¨of subset of X . On the other hand we have X \ Z ( f ) = S ∞ n =1 { x : | f ( x ) | ≥ n } , hence X \ Z ( f ) is a Lindel¨of subset of X ,by Theorem 3.8.5 in [10]. If X \ Z ( f ) be a Lindel¨of subset of X , then { x : | f ( x ) | ≥ n } is a Lindel¨of subset of X so { x : | f ( x ) | < n } contains an elementof P , i.e, f ∈ C ∞P ( X ). Similarly we may prove that C P ( X )= { f : X \ Z ( f ) isa Lindel¨of subset of X } .In the sequel, we assume that P is an open filter base for a filter F . Proposition 2.5.
If the complement of any element of P is Lindel¨of, then C ∞P ( X ) ⊆ T p ∈ νX \ X M p . Where by νX we mean the real compactification of X (see [11]). Proof.
Let f ∈ C ∞P ( X ) and p ∈ νX \ X (i.e., M p is a free real maximalideal). Then for any x ∈ X \ Z ( f ) there exist f x ∈ M P such that x ∈ X \ Z ( f x ). Hence X \ Z ( f ) ⊆ S x ∈ X X \ Z ( f x ). Now, by Example 2.4 andhypothesis, X \ Z ( f ) is Lindelof, so there is a countable subset S of X suchthat X \ Z ( f ) ⊆ S x ∈ S X \ Z ( f x ) and each f x ∈ M p . On the other hand,there exists h ∈ C ( X ) such that Z ( h ) = T x ∈ S Z ( f x ). Therefore h ∈ M p and Z ( h ) ⊆ Z ( f ). But M P is a z -ideal, so f ∈ M p , i.e., C ∞P ( X ) ⊆ T p ∈ νX \ X M p . (cid:3) Recall that, a subset A of X is called bounded (relative pseudocompact)subset, if for every function f ∈ C ( X ), f ( A ) is a bounded subset of R , see [14]. Example 2.6. P = { A : A is open and X \ A is pseudocompact } and X be anon-pseudocompact space. Then P is an open filter base and C P ( X ) = C ψ ( X ) = { f : X \ Z ( f ) is pseudocompact } , and C ∞P ( X )= { f : { x : | f ( x ) | ≥ n } is pseudocompact } .For, suppose that f ∈ C P ( X ), then Z ( f ) ⊇ A for some A ∈ P . Hence X \ Z ( f ) ⊆ X \ A . This implies that X \ Z ( f ) is a bounded subset. Now by[14, Theorem 2.1], X \ Z ( f ) is pseudocompact, i.e, f ∈ C ψ ( X ). If f ∈ C ψ ( X ),then X \ X \ Z ( f ) is an element of P and Z ( f ) ⊇ X \ X \ Z ( f ), i.e, f ∈ C P ( X ).For more details about C ψ ( X ), the reader referred to [14]. Similarly we mayprove the second equality. Remark 2.7.
Let P = { A : X \ A is finite } and X is an infinite space. Then C P ( X ) = C F ( X ), and C ∞P ( X ) = { f : { x : | f ( x ) | ≥ n } is finite for each n ∈ N } . In this case, C ∞P ( X ) = C F ( X ) if and only if the set of isolatedpoints of X is finite. To see this, let { x , x , ...x n , ... } be a subset of isolatedpoints in X . Define f n ( x ) = (cid:26) n x = x n x = x n and f ( x ) = P ∞ n =0 f n ( x ). Then ome generalizations and unifications of C K ( X ) , C ψ ( X ) and C ∞ ( X ) 6 f ∈ C ( X ), f ( x n ) = n and X \ Z ( f ) = { x , x , ... } , i.e., f / ∈ C F ( X ). On theother hand { x : | f ( x ) | < n } contains X \ { x , x , ..., x n } , so f ∈ C ∞P ( X ). Thisis a contradiction. Conversely, let f ∈ C ∞P ( X ) and f / ∈ C F ( X ). Then thereis { x , x , ...x n , ... } such that f ( x n ) = 0, so | f ( x n ) | > k n for some k n ∈ N , i.e., x n ∈ { x : | f ( x ) | ≥ k n } , but { x : | f ( x ) | > k n } is a finite open set so each x n isan isolated point, this is a contradiction. Definition 2.8.
A topological space X is called a local space provided that,there exists an open filter base P for a filter F on space X where F has nocluster point (i.e., T A ∈P A = ∅ ). Example 2.9.
Any locally compact non-compact space X is a local space.For see this, let P = { A ⊆ X : X \ A is compact } . Then if x ∈ T A ∈P A , bylocally compactness of X , there exists a compact subset K $ X such that x ∈ int ( K ), but X \ K is in P , hence x ∈ X \ K , i.e., int ( K ) ∩ ( X \ K ) = ∅ ,this is a contradiction, thus T A ∈P A = ∅ . Example 2.10. [11, 4. M] Let X be an uncountable space in which all pointsare isolated points except for a distinguished point s . A neighborhood of s isany set containing s which complement is countable. Then X is a local space.To see this, let Y = { x , x , ... } be a countable subset of X , where s / ∈ Y . Put A n = { x n , x n +1 , ... } and P = { A n : n ∈ N } . Then P is an open filter base on X .Now, for any n ∈ N , the set X \ A n is a neighborhood of s , so s / ∈ T A e n ∈P A n ,thus T A ∈P A = ∅ .In the following we see an example of a topological space which is not alocal space. Example 2.11. [11, 4. N] For each n ∈ N , let A n = { n, n +1 , ... } and E = { A n : n ∈ N } . Then E is a base for a free ultrafilter say E on N . Let X = N ∪ { σ } which points in N are isolated point and a neighborhood of σ is of the form U ∪ { σ } which U ∈ E . Note that any set contains σ is closed. Now if there isan open base P for some filter F on X such that F has no cluster point, thenthere exists F ∈ F such that σ / ∈ F , but σ has a neighborhood say U ∪ { σ } such that U ∈ E and U ∪{ σ } ⊆ X \ F . Since E is a base for E , then there exists n ∈ N and A n ∈ E such that A n ∪{ σ } = { n, n +1 , ... }∪{ σ } ⊆ U ∪{ σ } ⊆ X \ F .On the other hand, for points x = 1 , , ...n − F , F , ...F n − in F ,such that i ∈ X \ F i for 1 ≤ i ≤ n , so X ⊆ ( X \ F ) ∪ S n − i =1 ( X \ F i ), therefore ∅ ∈ F . This a contradiction, i.e., X is not a local space.We have already observe that C P ( X ) is a z -ideal in C ( X ). In the followingproposition we find a condition over which C P ( X ) is a free ideal. Proposition 2.12.
Let P be an open filter base for a filter F . Then C P ( X )is a free ideal if and only if F has no cluster point (i.e., T A ∈P A = ∅ ). ome generalizations and unifications of C K ( X ) , C ψ ( X ) and C ∞ ( X ) 7 Proof.
Let T F ∈F F = ∅ . Then there exists x ∈ T F ∈F F .
By hypothesis, thereis f ∈ C P ( X ) such that x ∈ X \ Z ( f ). On the other hand there is A ∈ F such that X \ Z ( f ) ⊆ X \ A . Hence ( X \ Z ( f )) ∩ A = ∅ . But x ∈ A impliesthat ( X \ Z ( f )) ∩ A = ∅ . This is a contradiction. Conversely, let x ∈ X . Wehave T F ∈F F = ∅ , so there exists F ∈ F such that x / ∈ F . By completelyregularity of X , there exists f ∈ C ( X ) such that f ( x ) = 1 and f ( F ) = 0.Hence f ∈ C P ( X ) and x / ∈ Z ( f ), i.e., C P ( X ) is a free ideal. (cid:3) It is easy to see that X is a locally compact non-compact space if andonly if P = { A ( X : X \ A is compact } is an open filter base for some filter F with no cluster point. So, by the above proposition we have the followingcorollaries. Corollary 2.13. C K ( X ) is a free ideal if and only if X is a locally compactnon-compact space. Corollary 2.14.
A space X is a local space if and only if C P ( X ) is a freeideal for some open filter base P on X . Proof.
By Proposition 2.12, the verification is immediate. (cid:3)
Proposition 2.15.
Let P be an open filter base. The following statementsare equivalent.1. Every element of P is dense in X .2. C ∞P ( X ) = (0) . C P ( X ) = (0) . Proof. (1) ⇒ (2) Let for every A ∈ P , A = X and f ∈ C ∞P ( X ). Then the set { x : | f | ≤ n } = X , so for any n > { x : | f | < n − } ⊇ { x : | f | ≤ n } = X , i.e., f = 0.(2) ⇒ (3) This is evident.(3) ⇒ (1) Suppose that C P ( X ) = 0 and A ∈ P . If A = X , then there exists x ∈ X \ A , hence we define f ∈ C ( X ) such that f ( x ) = 1 and f ( A ) = 0, i.e., f ∈ C P ( X ) = 0, which implies that f = 0, this is a contradiction. (cid:3) Corollary 2.16.
Let X = Q with usual topology and P = { A ⊂ Q : Q \ A iscompact } . Then C ∞P ( X ) = C ∞ ( X ) = (0). Proof.
Every element of P is dense in X , so by Proposition 2.15, C ∞P ( X ) = C ∞ ( X ) = (0). (cid:3) Definition 2.17.
Let F be a filter on X . A subset A of X is an F - CG δ if A = T ∞ i =1 A i , where each A i ∈ F and is open and for each i , X \ A i and A i +1 are completely separated (see [11]). ome generalizations and unifications of C K ( X ) , C ψ ( X ) and C ∞ ( X ) 8 Example 2.18.
Let F = { F $ X : X \ F is compact } and X is a non-compactHausdorff space. Then for every open locally compact σ -compact subspace A , X \ A is an F - CG δ . Since, by [10, 3. p. 250], A = S ∞ i =1 A i where A i ⊆ intA i +1 and each A i is compact so X \ A is an F - CG δ set.In the following lemma, we give a characterization of a closed F - CG δ sub-set. Lemma 2.19.
Let A be a closed subset of space X . Then A is an F - CG δ setif and only if A = Z ( f ) for some f ∈ C ∞P ( X ). Proof.
Let A be an F - CG δ . Then A = T ∞ n =1 A i , where each A n is a an elementof F , X \ A n and A n +1 are completely separated. Now for each n ∈ N , thereexists f n ∈ C ( X ) such that f n ( A n +1 ) = 0, f n ( X \ A n ) = 1, then f = P n f n isan element of C ( X ), by Weierstrass M-test. Clearly A = Z ( f ). We claim that f ∈ C ∞P ( X ). Let x ∈ A n +1 . Then f ( x ) = f ( x ) = ...f n ( x ) = 0 and so f ( x ) ≤ n +1 + n +1 + ... ≤ n < n . Therefore x ∈ { x : | f ( x ) < n } , and hence A n +1 ⊆ { x : | f ( x ) < n } , i.e, f ∈ C ∞P ( X ). Conversely, suppose that A = Z ( f )for some f ∈ C ∞P ( X ). Then A = T ∞ n =1 A n , where A n = { x : | f ( x ) | < n } ∈ F for each n ∈ N , X \ A n and A n +1 are disjoint zero-sets, and hence completelyseparated, i.e., A is an F - CG δ . (cid:3) Proposition 2.20. C ∞P ( X ) = C P ( X ) if and only if every closed F - CG δ isan element of F . Proof.
Suppose that condition holds. We know that C P ( X ) is a subset of C ∞P ( X ). It is enough to prove that C ∞P ( X ) ⊆ C P ( X ). Let f ∈ C ∞P ( X ),then by Lemma 2.12, Z ( f ) is a closed F - CG δ . Hence Z ( f ) contains an elementof P , i.e, f ∈ C P ( X ). Conversely, suppose that C ∞P ( X ) = C P ( X ) and A is a closed F - CG δ . By lemma 2.12, A = Z ( f ) for some f ∈ C ∞P ( X ), now f ∈ C P ( X ) implies that A = Z ( f ) contains an element of P , i.e, A ∈ F . (cid:3) In the above proposition we have seen that if every closed F - CG δ is anelement of F , then C ∞P ( X ) is an ideal of C ( X ). But in general, C ∞P ( X ) maynot be an ideal of C ( X ) as we will see in the sequel. Example 2.21.
Let P = { R \ [ n , n ]: n ∈ N } . Then it is easy to see that P isan open filter base on R . Now, we show that C ∞P ( R ) $ C ∞ ( R ) is not an idealof C ( R ). For see this, Consider f ( x ) = x ≤ x ≤ x ≤ x ≤ x , g ( x ) = x ≤ x ≤ x ≤ x ≤ x . Then f ∈ C ∞P ( R ), g ∈ C ( R ) and we have ome generalizations and unifications of C K ( X ) , C ψ ( X ) and C ∞ ( X ) 9( f g )( x ) = x ≤ x ≤ x ≤
11 1 ≤ x But f g / ∈ C ∞P ( R ), because { x : | ( f g )( x ) | < } =( −∞ , x +1 ∈ C ∞ ( R ) and is not in C ∞P ( R ), so C ∞P ( R ) $ C ∞ ( R ).In the following, we provide an example of open filter base P for which C ∞P ( X ) is an ideal of C ( X ). Example 2.22. P = { A : A is open and X \ A is Lindel¨of } and X be a non-Lindel¨of space. Then by Example 2.4, C ∞P ( X )= { f : X \ Z ( f ) is Lindel¨ofsubset of X } , which is an ideal of C ( X ). For, let f, g ∈ C ∞P ( X ), then X \ Z ( f + g ) ⊆ ( X \ Z ( f )) ∪ ( X \ Z ( g )) implies { x : | ( f + g )( x ) | < n } ⊇ X \ (( X \ Z ( f )) ∪ ( X \ Z ( g )), which contains an element of P , i.e, f + g ∈ C ∞P ( X ).If f ∈ C ∞P ( X ) and g ∈ C ( X ), then X \ Z ( f g ) ⊆ X \ Z ( f ) implies that { x : | ( f.g )( x ) | < n } contains an element of P , i.e, f.g ∈ C ∞P ( X ).Azarpanah and Soundarajan in [6] have found some equivalent conditionsfor which C ∞ ( X ) is an ideal of C ( X ) (i.e., P = { A $ X : X \ A is compact } ).In the following theorem, we find some equivalent conditions for a larger classof P which C ∞P ( X ) is an ideal of C ( X ) and by this theorem give severalcorollary. Theorem 2.23.
The following statements are equivalent.1. C ∞P ( X ) is an ideal of C ( X ) and for any F ∈ F , X \ F is bounded.2. The set of non-cluster points of filter F is bounded.3. The complement of every closed F - CG δ is bounded. Proof. (1) ⇒ (2). Let A be the set of non-cluster points of filter F . If A is not bounded, then there exist h ∈ C ( X ) and a discrete subset C = { x , x , x , ... } ⊆ A such that for each n ∈ N , h ( x n ) ≥ n . A is open soany { x n } is an isolated point, therefore we can define f n ( x ) = (cid:26) n x = n x = n and f ( x ) = P n = ∞ n =0 f n ( x ) such that f n ∈ C ( X ) and so f ∈ C ( X ). We have { x : | f | < n } = { x n +1 , x n +2 , ... } . Now any x n ∈ X \ F n for some F n ∈ F .This implies that X \ { x , ..., x n } contains an element of F hence contains anelement of P , i.e., f ∈ C ∞P ( X ). But we have { x : | f h | < n } = X \ { x , x ..., } ,if there is P ∈ P such that X \ { x , x ..., } ⊇ P , then { x , x , ... } ⊆ X \ P ,which contradict by hypothesis, so f h / ∈ C ∞P ( X ), i.e., C ∞P ( X ) is not an idealof C ( X ), this is a contradiction.(2) ⇒ (3). It is easily seen that the complement of every closed F - CG δ is asubset of non-cluster points of the filter F so is bounded. ome generalizations and unifications of C K ( X ) , C ψ ( X ) and C ∞ ( X ) 10(3) ⇒ (1). By Lemma 2.1, C ∞P ( X ) is a subring of C ( X ), it is enough toprove that f g ∈ C ∞P ( X ) for every f ∈ C ( X ) and g ∈ C ∞P ( X ). By Lemma2.12, Z ( g ) is an F - CG δ and hence, by (3), Y = X \ Z ( g ) is a bounded subset of X , so f ( Y ) is a bounded subset of R . Now it is easy to see that g ∈ C ∞P ( X ),since g ∈ C ∞P ( X ), moreover ( f ( g ) )( X ) = ( f ( g ) )( Y ) ∪ { } . Since f ( Y ) is abounded subset of R and g ∈ C ∞P ( X ) is a bounded function on X (note that X \ Z ( g ) is bounded), we get ( f ( g ) )( Y ) is a bounded set in R , this implies that( f ( g ) ) is a bonded on X and so belong to C ∗ ( X ). Since C ∞P ( X ) is a ring, g ∈ C ∞P ( X ). However, since for any f ∈ C ∞P ( X ), X \ Z ( f ) is bounded, then C ∞P ( X ) is an ideal of C ∗ ( X ). Therefore f g = ( f ( g ) )( g ) ∈ C ∞P ( X ), thus C ∞P ( X ) is an ideal of C ( X ). Now let F ∈ F and X \ F is unbounded. Thenit contains infinity set of isolated points say A = { x , x , ... } ⊆ ( X \ F ), suchthat A is unbounded. We have X \ A = T ∞ i =1 B i where B i = X \ { x , ..., x i } .Clearly for each i B i ∈ F and X \ B i , B i +1 are completely separated. Hence X \ A is an F - CG δ , by hypothesis, X \ ( X \ A ) = A is bounded, this is acontradiction. This completes the proof. (cid:3) Corollary 2.24. If X is a pseudocompact space, then for any open filter base P , C ∞P ( X ) is an ideal of C ( X ). Proof. If X is a completely regular pseudocompact Hausdorff space and P bean open filter base for filter F , then any subset of X is bounded so the set ofnon-cluster point of F is bounded, thus by Theorem 2.23, C ∞P ( X ) is an idealof C ( X ). (cid:3) Corollary 2.25.
Let X be a local space. Then for any open filter base P , C ∞P ( X ) is an ideal of C ( X ) and for any A ∈ P , X \ A is bounded if and onlyif X is a pseudocompact non-compact space. Proof. If X is pseudocompact, then by Corollary 2.24, for any open filter base P , C ∞P ( X ) is an ideal of C ( X ) and for any A ∈ P , X \ A is bounded. Nowlet X be a local space, then there exist an open filter base P for some filter F on X such that F has no cluster point so X is the set of non-cluster point offilter F . Hence by Theorem 2.23, X is bounded, i.e., X is pseudocompact. (cid:3) Corollary 2.26.
Let X be a non-pseudocompact space and P = { A : A is openand X \ A is bounded } . Then C ∞P ( X ) is an ideal of C ( X ) if and only if anyunion of the interior of closed bounded subsets is a bounded subset. Proof.
If any union of the interior of closed bounded subsets is a boundedsubset, then the set of non-cluster points of open filter P is bounded so byTheorem 2.23, C ∞P ( X ) is an ideal of C ( X ). Conversely, If A = S α ∈ S int A α where for each α ∈ S , A α is a closed bounded set, then we have X \ A α ∈ P and int A α = X \ ( X \ A α ), so A is contained in the set of non-cluster pointsof open filter P , i.e., A is bounded. (cid:3) ome generalizations and unifications of C K ( X ) , C ψ ( X ) and C ∞ ( X ) 11 Corollary 2.27.
Let X be an an infinite space and P = { A ( X : X \ A isfinite } . Then C ∞P ( X ) is an ideal of C ( X ) if and only if the set of isolatedpoints of X is bounded. Proof.
Let A ∈ P . We have X \ A is an open finite subset, thus the set ofnon-cluster points of P is contained in the set of isolated points of X , so isbounded, hence by Theorem 2.23, C ∞P ( X ) is an ideal of C ( X ). Conversely,let A be the set of isolated points of X . Then A = S x ∈ A { x } , each { x } is aclopen subset, X \{ x } ∈ P and { x } =int { x } = X \ ( X \ { x } ), so A is containedin the set of non-cluster points of open filter P , i.e, A is bounded. (cid:3) Example 2.28. (a). If P = { A ( R : R \ A is bounded } . Then C ∞P ( R ) is notan ideal of C ( R ). Because S ∞ n =1 (0 , n ) is not bounded.(b). If P = { A ( R : R \ A is finite } . Then by Corollary 2.27, C ∞P ( R ) isan ideal of C ( R ). Remark 2.29.
Any closed bounded in a normal space is a pseudocompact andany pseudocompact Lindel¨of space is compact, so if X is a realcompact normalspace and P equal be the set of all subsets whose complements are boundedsubsets of X , then C ∞P ( X ) = C ∞ ( X ), for example in R if P equal be the setof all subsets of R , whose complements are bounded, then C ∞P ( R ) = C ∞ ( R ).In particularly, if X is a Lindel¨of space and P equal be the set of all subsetsof X , whose complements are bounded, then C ∞P ( X ) = C ∞ ( X ). C P ( X ) as an essential ideal. Topological spaces X for which C ∞ ( X ) (resp., C K ( X )) is an essential ideal wascharacterized by Azarpanah, in [4]. In this section we characterize topologicalspaces X for which C P ( X ) is an essential ideal. Proposition 3.1.
An ideal E in C ∞P ( X ) is an essential ideal if and only if T Z [ E ] does not contain a subset V where X \ V ∈ F and intV = ∅ . Proof.
Let X \ V = F ∈ F , S f ∈ E coz ( f ) ⊆ F and intV = ∅ , i.e., F = X .Then there exist x ∈ X such that x / ∈ F , it follows that there is f ∈ C ( X )such that f ( x ) = 1, f ( F ) = 0, i.e., f ∈ C ∞P ( X ). Now for any g ∈ E we have X \ Z ( f ) ⊆ X \ F ⊆ Z ( g ), i.e., f g = 0 so ( f ) ∩ E = 0, which contradictsthe essentiality of E . Conversely, let 0 = f ∈ C ∞P ( X ). Then there is a ∈ X such that | f ( a ) | > n for some n ∈ N , hence a ∈ X \ { x : | f | ≤ n } , i.e., { x : | f | ≤ n } 6 = X . We know that { x : | f | < n } ∈ F . By hypothesis, X \ { x : | f | < n } * T Z [ E ]. Therefore, there exists b ∈ X \ { x : | f | ≤ n } and g ∈ E such that g ( b ) = 0, i.e, f g = 0 thus E is an essential ideal in C ∞P ( X ). (cid:3) ome generalizations and unifications of C K ( X ) , C ψ ( X ) and C ∞ ( X ) 12Recall that a collection B of open sets in a topological space X is called a π -base if every open set of X contains a member of B . The reader is referredto [8], [9], [10], [12], and [15]. The next result is a generalization of Theorem3.2 in [4]. Theorem 3.2. C P ( X ) is an essential ideal if and only if { V : V is open and X \ V ∈ F } is a π -base for X . Proof.
Let U be a proper open set in X . By regularity of X , there exist anon-empty open set V such hat V ⊆ clV ⊆ U . Now find f ∈ C ( X ) where f ( clV ) = { } , f ( x ) = 0 for some x / ∈ U . If X \ V ∈ F , there is notingto proved. Suppose X \ V / ∈ F . If V ⊆ Z ( h ) for every h ∈ C P ( X ), then V ⊆ T Z [ C P ( X )], which implies that C P ( X ) is not an essential ideal, by [5,Theorem 3.1]. Therefor there is some h ∈ C P ( X ) such that V ∩ ( X \ Z ( h )) = ∅ ,i.e, there is some x ∈ V for which h ( x ) = 0. Clearly f h ∈ C P ( X ). So W = X \ Z ( f h ) is contained in X \ F for some F ∈ F . If W ′ = W ∩ V , then W ′ is a non-empty open set in U and X \ W ′ ∈ F .Conversely, We will prove that for every non-unit g ∈ C ( X ), C P ( X ) ∩ ( g ) = 0. Since X \ Z ( g ) is an open set, then there is an open set U where U ⊆ clU ⊆ X \ Z ( g ), and there is an open set V ⊆ U such that X \ V ∈ F .Then V ⊆ U ⊆ X \ Z ( g ). Define f ∈ C ( X ) such that f ( X \ V ) = 0 , f ( x ) = 1for some x ∈ V . Since X \ V ⊆ Z ( f ) so f ∈ C P ( X ) . On the other hand Z ( g ) ⊆ X \ V ⊆ Z ( f ) so f g = 0 and f g ∈ C P ( X ) ∩ ( g ). (cid:3) C ∞P ( X ) as a z -ideal and a regular ring. We know that C ∞P ( X ) is a subring of C ( X ), in this section, we see that C ∞P ( X ) is a z -ideal if and only if every cozero-set containing a F - CG δ is anelement of F . Also we prove that, C ∞P ( X ) is a regular ring (i.e., for each f ∈ C ∞P ( X ) there exists g ∈ C ∞P ( X ) such that f = f g ) if and only if everyclosed F - CG δ is an open subset and belong to F . Proposition 4.1.
The subring C ∞P ( X ) is a z -ideal of C ( X ) if and only ifevery cozero-set containing a closed F - CG δ is an element of F . Proof.
First, we prove that Z ( f ) ⊆ Z ( g ) and f ∈ C ∞P ( X ), implies that g ∈ C ∞P ( X ). To see this, we know that Z ( f ) ⊆ Z ( g ) ⊆ { x : | g ( x ) | < n } , forall n ∈ N . But { x : | g ( x ) | < n } is a cozero-set and Z ( f ) is a closed F - CG δ .So, by hypothesis, { x : | g ( x ) | < n } is an element of F , i.e., g ∈ C ∞P ( X ). Now,suppose that f ∈ C ∞P ( X ) and g ∈ C ( X ). Then Z ( f ) ⊆ Z ( f g ), shows that f g ∈ C ∞P ( X ). Thus C ∞P ( X ) is a z -ideal of C ( X ). Conversely, Suppose that X \ Z ( f ) is a cozero-set contains a closed F CG δ subset A . By Lemma 2.19,there exist g ∈ C ∞P ( X ), such that A = Z ( g ), so Z ( g ) ⊆ X \ Z ( f ). Now we ome generalizations and unifications of C K ( X ) , C ψ ( X ) and C ∞ ( X ) 13define h = g f + g . Then h ∈ C ( X ) and Z ( g ) ⊆ Z ( h ), therefore h ∈ C ∞P ( X ).On the other hand, for each n ∈ N , { x : | h ( x ) | < n } ⊆ X \ Z ( f ), hence X \ Z ( f ) ∈ F . This completes the proof. (cid:3) Theorem 4.2. C ∞P ( X ) is a regular ring if and only if every closed F - CG δ is an open subset and belongs to F . Proof.
First, we prove that every closed F - CG δ is an open subset. By Lemma2.19, every closed F - CG δ is of the form Z ( f ) for some f ∈ C ∞P ( X ). But Z ( f ) = Z ( f ∧ n ), for each n ∈ N and { x : | f ∧ n | < m } = { x : | f | < m } . So we can let f is bounded. Regularity of C ∞P ( X ) implies that, thereexists g ∈ C ∞P ( X ) such that f = f g . Then X \ Z (1 − f g ) ⊆ intZ ( f ).If x ∈ Z ( f ) \ intZ ( f ), then x ∈ Z (1 − f g ), which contradict x ∈ Z ( f ),i.e., Z ( f ) is an open subset. On the other hand for every x ∈ X \ Z ( f ), g ( x ) = f ( x ) and hence g ( x ) ≥ n , where n is an upper bounded for | f | . Therefore X \ Z ( f ) ⊆ { x : | g | ≥ n } , i.e., Z ( f ) ⊇ { x : | g | < n } . But { x : | g | < n } containsan element of P so Z ( f ) ∈ F . Conversely, Suppose f ∈ C ∞P ( X ). Z ( f ) isa closed F - CG δ so by hypothesis, is an open subset which belong to F . Wedefine g ( x ) = 0 for x ∈ Z ( f ) and g ( x ) = f ( x ) for x ∈ X \ Z ( f ). Then g ∈ C ( X ), f = f g and { x : | g | < n } ⊇ Z ( f ). , i.e., g ∈ C ∞P ( X ). (cid:3) Corollary 4.3. (a) Let P = { A : A is open and X \ A is Lindel¨of } and X is anon-Lindel¨of space. Then P is an open filter and C ∞P ( X ) is a regular ring ifand only if every closed P - CG δ is an open subset.(b) C ∞ ( X ) is a regular ring if and only if every open locally compact σ -compact subset is compact. Proof. (a) It is easily seen that P is an open filter. If C ∞P ( X ) is a regularring, then by Theorem 4.2, every closed P - CG δ is an open subset. Now let A be a closed P - CG δ which is an open subset. Then by Lemma 2.19, A = Z ( f )for some f ∈ C ∞P ( X ). But X \ A = X \ Z ( f ) = S ∞ n =1 { x : | f ( x ) | ≥ n } , henceby [10, Theorem 3.8.5], X \ A is a Lindel¨of subset of X , i.e, A ∈ P . ByTheorem 4.2, C ∞P ( X ) is a regular ring.(b) If A is an open locally compact σ -compact subset, then X \ A is a closed P - CG δ , where P = { A ( X : X \ A is compact } and C ∞ ( X ) = C ∞P ( X ), so byTheorem 4.2, we are done. (cid:3) ACKN OW LEDGEM EN T S
The author would like to thank Prof. Momtahen for his encouragement anddiscussion on this paper. ome generalizations and unifications of C K ( X ) , C ψ ( X ) and C ∞ ( X ) 14 References [1] S. K. Acharyya and S. K. Ghosh,
Functions in C ( X ) with support lyingon a class of subsets of X , Topology procedding , 35(2010) , Rings of continuous functionsvanishing at infinity , Comment. Math. Univ. Carolinae
Relative z -ideals in com-mutative rings , comm. Algebra. Intersection of essential ideals in C ( X ), Proc. Amer. Math.Soc. , 125(1997) , Essential ideals in C ( X ), Period. Math. Hungar. ,31(1995) , When the family of functions van-ishing at infinity is an ideal of C ( X ), Rocky Mountain, journal of math-ematics , 31(4) , , z -ideals in C( X ). Topology andits applications . 156(2009)1711–1717.[8] A. Bella, A. W. Hager, J. Martinez, S. Woodward, H. Zhou,
Speakerspaces and their absolutes, I. Top. Appl. , – . [9] A. Bella, J. Martinez, S. Woodward, Algebra and spaces of dense constan-cies, Czechoslovak Math. J. , – . [10] R. Engelking, General Topology , PWN-Polish Sci. Publ (1977).[11] L. Gilman, and M. jerison,
Ring of Continuous Function , Springer-verlag(1976).[12] M. L. Knox, and W. Wm. McGovern,
Rigid extensions of l -groups ofcontinuous functions, Czechoslovak Math. J. , – . [13] C. W. Kohls, Ideals in rings of continuous functions, Fund. Math. , Supports of continuous functions , Trans. Amer. Math. Soc. , Clean semiprime f -rings with bounded inversion,Comm. Algebra. , –3304