@ Appl. Gen. Topol. 20, no. 2 (2019), 395-405 doi:10.4995/agt.2019.11524 c© AGT, UPV, 2019 ec-Filters and ec-ideals in the functionally countable subalgebra of C∗(X) Amir Veisi Faculty of Petroleum and Gas, Yasouj University, Gachsaran, Iran (aveisi@yu.ac.ir) Communicated by A. Tamariz-Mascarúa Abstract The purpose of this article is to study and investigate ec-filters on X and ec-ideals in C ∗ c (X) in which they are in fact the counterparts of zc-filters on X and zc-ideals in Cc(X) respectively. We show that the maximal ideals of C∗c (X) are in one-to-one correspondence with the ec-ultrafilters on X. In addition, the sets of ec-ultrafilters and zc- ultrafilters are in one-to-one correspondence. It is also shown that the sets of maximal ideals of Cc(X) and C ∗ c (X) have the same cardinality. As another application of the new concepts, we characterized maximal ideals of C∗c (X). Finally, we show that whether the space X is compact, a proper ideal I of Cc(X) is an ec-ideal if and only if it is a closed ideal in Cc(X) if and only if it is an intersection of maximal ideals of Cc(X). 2010 MSC: 54C30; 54C40; 54C05; 54G12; 13C11; 16H20. Keywords: c-completely regular space; closed ideal; functionally countable space; ec-filter; ec-ideal; zero-dimensional space. 1. Introduction All topological spaces are completely regular Hausdorff spaces and we shall assume that the reader is familiar with the terminology and basic results of [6]. Given a topological space X, we let C(X) denote the ring of all real-valued continuous functions defined on X. Cc(X) is the subalgebra of C(X) consisting of functions with countable image and C∗c (X) is its subalgebra consisting of bounded functions. In fact, C∗c (X) = Cc(X)∩C∗(X), where elements of C∗(X) Received 19 March 2019 – Accepted 30 July 2019 http://dx.doi.org/10.4995/agt.2019.11524 A. Veisi are bounded functions of C(X). Recall that for f ∈ C(X),Z(f) denotes its zero-set: Z(f) = {x ∈ X : f(x) = 0}. The set-theoretic complement of a zero-set is known as a cozero-set and we denote this set by coz(f). Let us put Zc(X) = {Z(f) : f ∈ Cc(X)} and Z∗c (X) = {Z(g) : g ∈ C∗c (X)}. These two latter sets are in fact equal, since Z(f) = Z( f 1+|f|), where f ∈ Cc(X). A nonempty subfamily F of Zc(X) is called a zc-filter if it is a filter on X. If I is an ideal in Cc(X) and F is a zc-filter on X then, we denote Zc[I] = {Z(f) : f ∈ I}, ∩Zc[I] = ∩{Z(f) : f ∈ I} and Z−1c [F] = {f : Z(f) ∈F}. We see that Zc[I] is a zc-filter and Z−1c [Zc[I]] ⊇ I. If the equality holds, then I is called a zc-ideal. Moreover, Z −1 c [F] is a zc-ideal and we always have Zc[Z −1 c [F]] = F. So maximal ideals in Cc(X) are zc-ideals. In [5], a Huasdorff space X is called countably completely regular (briefly, c- completely regular) if whenever F is a closed subset of X and x /∈ F, there exists f ∈ Cc(X) such that f(x) = 0 and f(F) = 1. In addition, two closed sets A and B of X are also called countably separated (in brief, c-separated) if there exists f ∈ Cc(X) with f(A) = 0 and f(B) = 1. c-completely regular and zero-dimensional spaces are the same, see Theorem 1.1. If we let Mcp = {f ∈ Cc(X) : f(p) = 0} (p ∈ X), then the ring isomorphism Cc(X) Mcp ∼= R gives that Mcp is a maximal ideal, in fact, Mcp is a fixed maximal ideal. Moreover, ∩Zc[Mcp] = {p}. Our concentration is on the zero-dimensional spaces since in [5] the authors proved that for any space X there is a zero-dimensional Hausdorff space Y such that Cc(X) and Cc(Y ) are isomorphic as rings, see Theorem 1.2. In section 2, we study and investigate the ec-filters on X and ec-ideals in C∗c (X) which they are in fact the counterpart of [6, 2L]. We show that the max- imal ideals of C∗c (X) are in one-to-one correspondence with the ec-ultrafilters on X. Moreover, the sets of ec-ultrafilters and zc-ultrafilters are in one-to-one correspondence. By using the latter facts, it is shown that the sets of maximal ideals of Cc(X) and C ∗ c (X) have the same cardinality. Finally, maximal ideals of C∗c (X) are characterized based on these concepts. In Section 3, our concen- tration is on the uniform norm topology on C∗c (X) which is the restriction of the uniform norm topology on C∗(X). It is shown that whenever the space X is compact, a proper ideal I of Cc(X) is an ec-ideal if and only if it is a closed ideal in Cc(X) if and only if it is an intersection of maximal ideals of Cc(X). We recite the following results from [5]. Theorem 1.1 ([5, Proposition 4.4]). Let X be a topological space. Then, X is a zero-dimensional space (i.e., a T1-space with a base consisting of clopen sets) if and only if X is c-completely regular space. Theorem 1.2 ([5, Theorem 4.6]). Let X be any topological space (not neces- sarily completely regular). Then, there is a zero-dimensional space Y which is a continuous image of X with Cc(X) ∼= Cc(Y ) and CF (X) ∼= CF (Y ). c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 2 396 ec-Filters and ec-ideals in the functionally countable subalgebra of C ∗(X) Remark 1.3 ([5, Remark 7.5]). There is a topological space X, such that there is no space Y with Cc(X) ∼= C(Y ). The following results are the known facts about Cc(X) and we are seeking to get similar results for C∗c (X). Proposition 1.4. Let I be a proper ideal in Cc(X) and F a zc-filter on X. Then: (i) Zc[I] is a zc-filter and Z −1 c [F] is a zc-ideal of Cc(X). (ii) If I is maximal then Zc[I] is a zc-ultrafilter, and the converse holds if I is a zc-ideal. (iii) F is a zc-ultrafilter if and only if Z−1c [F] is a maximal ideal. (iv) If F is a zc-ultrafilter and Z ∈ Zc(X) meets each element of F, then Z ∈F. Corollary 1.5. There is a one-to-one correspondence ψ between the sets of zc-ideals of Cc(X) and zc-filters on X, defined by ψ(I) = Zc[I]. In particular, the restriction of ψ to the set of maximal ideals is a one-to-one correspondence between the sets of maximal ideals of Cc(X) and zc-ultrafilters on X. 2. ec-filters on X and ec-ideals in C ∗ c (X) For f ∈ C∗c (X) and � > 0, we define Ec� (f) = f −1([−�,�]) = {x ∈ X : |f(x)| ≤ �}. Each such set is a zero set, since it is equal to Z((|f|−�)∨0). Conversely, every zero set is also of this form, since for g ∈ C∗c (X) we have Z(g) = Ec� (|g| + �). For a nonempty subset I of C∗c (X) we denote E c � [I] = {Ec� (f) : f ∈ I}, and Ec(I) = ⋃ � E c � [I]. Moreover, if F is a nonempty subfamily of Z∗c (X), then we define Ec� −1[F] = {f ∈ C∗c (X) : Ec� (f) ∈F} and E−1c (F) = ⋂ � E c � −1[F]. So we have Ec(I) = {Ec� (f) : f ∈ I and � > 0}, and E−1c (F) = {f ∈ C∗c (X) : Ec� (f) ∈ F, for all �}. Moreover, E−1c (Ec(I)) = {g ∈ C∗c (X) : Ecδ(g) ∈ Ec(I), for all δ > 0} and Ec(E−1c (F)) = {Ec� (f) : Ecδ(f) ∈F, for all δ > 0}. The next result is now immediate. Corollary 2.1. The following statements hold. (i) I ⊆ E−1c (Ec(I)) and Ec(E−1c (F)) ⊆F. (ii) The mappings Ec and E −1 c preserve the inclusion. (iii) If f ∈ I then for each positive integer n, Ec� (f) = Ec�n(fn). (iv) If I is an ideal, then Ec(I) is a zc-filter. Proof. The proofs of (i), (ii) and (iii) are clear. (iv). This is presented in the proof of Proposition 2.5. � Examples 2.2 and 2.3 below show that the inclusions in (i) of the above corollary may be strict even when I is an ideal and F is a zc-filter. c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 2 397 A. Veisi Example 2.2. Let X be the discrete space N×N, f(m,n) = 1 mn and I the ideal in C∗c (X)(= C ∗(X)) generated by f2. Obviously f /∈ I. Since {x ∈ X : f(x) ≤ �} = {x ∈ X : f2(x) ≤ �2}, we have Ec� (f) ∈ Ec(I). So I $ E−1c (Ec(I)). Example 2.3. Let X be the zero-dimensional space Q×Q, where Q is the set of rational numbers, and F = {Z ∈ Zc(X) : (0, 0) ∈ Z}. Then F is a zc-filter on X. Now, if we define f(x,y) = |x|+|y| 1+|x|+|y| then f ∈ C ∗ c (X)(= C ∗(X)) and Z(f) = {(0, 0)}. Given � > 0 and g = f + �, so we have Ec� (g) = {(0, 0)}. If we take 0 < δ < � then Ecδ(g) = ∅. Hence E c � (g) is not contained in Ec(E −1 c (F)). Therefore the latter set is contained in F properly, which gives the result. Definition 2.4. A zc-filter F is called an ec-filter if F = Ec(E−1c (F)), or equivalently, whenever Z ∈F then there exist f ∈ C∗c (X) and � > 0 such that Z = Ec� (f) and E c δ(f) ∈F, for each δ > 0. Proposition 2.5. If I is a proper ideal in C∗c (X), then Ec(I) is an ec-filter. Proof. First, we show that Ec(I) is a zc-filter, i.e., it satisfies the following conditions. (i) ∅ /∈ Ec(I). (ii) Ec� (f),E c δ(g) ∈ Ec(I), then E c � (f) ∩Ecδ(g) ∈ Ec(I). (iii) Ec� (f) ∈ Ec(I), Z ∈ Zc(X) with Z ⊇ Ec� (f), then Z ∈ Ec(I). (i). Suppose that for some � > 0 and f ∈ I, Ec� (f) = ∅. So � < |f|, which yields f is a bounded away from zero. Hence I contains the unit f, which is impossible. (ii). This is equivalent to say that if Ec� (f),E c δ(g) ∈ Ec(I), then Ec� (f) ∩ Ecδ(g) contains a member of Ec(I). Suppose that f ′,g′ ∈ I and �′,δ′ > 0 such that Ec� (f) = E c �′(f ′) and Ecδ(g) = E c δ′(g ′). Without loss of generality, we may suppose that δ′ < �′ < 1. Hence f′2 + g′2 ∈ I and Ec δ′2 (f′2 + g′2) ⊆ Ec� (f) ∩ Ecδ(g), which gives the result. (iii). Assume that Ec� (f) ⊆ Z(f′), where f ∈ I and f′ ∈ C∗c (X). Since Ec� (f) = Ec�2 (f 2) and Z(f′) = Z(|f′|), we can suppose that f ≥ 0 and f′ ≥ 0. Now, define g(x) = { 1, if x ∈ Ec� (f) f′(x) + � f(x) , if x ∈ X r intEc� (f). So g is continuous, since it is continuous on two closed sets whose union is X, in fact, g ∈ C∗c (X). Note that fg ∈ I and (fg)(x) = { f(x), if x ∈ Ec� (f) (ff′)(x) + �, if x ∈ X r intEc� (f). It is easily seen that Z(f′) = Ec� (fg). So Z(f ′) ∈ Ec(I). This shows that Ec(I) is a zc-filter. Now, apply (i) and (ii) of Corollary 2.1 for the ideal I and the zc-filter Ec(I), to get the inclusions Ec(I) ⊆ Ec(E−1c (Ec(I))) and Ec(E −1 c (Ec(I))) ⊆ Ec(I), which yields Ec(I) is an ec-filter. � Definition 2.6. An ideal I in C∗c (X) is called ec-ideal if I = E −1 c (Ec(I)), or equivalently, if f ∈ C∗c (X) and Ec� (f) ∈ Ec(I) for all �, then f ∈ I. c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 2 398 ec-Filters and ec-ideals in the functionally countable subalgebra of C ∗(X) Proposition 2.7. If F is a zc-filter, then E−1c (F) is an ec-ideal in C∗c (X). Proof. Let f,g ∈ E−1c (F), h ∈ C∗c (X) and let u be an upper bound for h and � > 0. Then Ec� 2 (f), Ec� 2 (g) and hence Ec� 2 (f) ∩ Ec� 2 (g) belong to F. Hence Ec� 2 (f)∩Ec� 2 (g) ⊆ Ec� (f +g) implies that Ec� (f +g) ∈F, or equivalently, f +g ∈ E−1c (F). Moreover, Ec� u (f) ⊆ Ec� (fh) implies fh ∈ E−1c (F). Therefore E−1c (F) is ideal. In view of Corollary 2.1, we have E−1c (F) ⊆ E−1c (Ec(E−1c (F))) ⊆ E−1c (F) and so the equality holds, i.e., E−1c (F) is an ec-ideal. � Corollary 2.8. Maximal ideals of C∗c (X) and an arbitrary intersection of them are ec-ideals. Proof. Let M be a maximal ideal of C∗c (X). If E −1 c (Ec(M)) is not a proper ec-ideal, then it contains the constant function 1 and E c � (1) = ∅ ∈ Ec(M) (0 < � < 1) which is impossible, see Propositions 2.5 and 2.7. Hence M = E−1c (Ec(M)), i.e., M is an ec-ideal. The second part is obtained by this fact, the fact that the intersection of a family of maximal ideals is an ideal contained in each of them and (ii) of Corollary 2.1. � The next corollary is an immediate result of Propositions 2.5 and 2.7. Corollary 2.9. The correspondence I 7→ Ec(I) is one-one from the set of ec-ideals in C ∗ c (X) onto the set of ec-filters on X. Lemma 2.10. (i) Let I and J be ideals in C∗c (X) and J an ec-ideal. Then I ⊆ J if and only if Ec(I) ⊆ Ec(J). (ii) Let F1 and F2 be zc-filters on X and F1 an ec-filter. Then F1 ⊆F2 if and only if E−1c (F1) ⊆ E−1c (F2). Proof. It is straightforward. � Proposition 2.11. Let I be an ideal in C∗c (X) and F a zc-filter on X. Then: (i) E−1c (Ec(I)) is the smallest ec-ideal containing I. (ii) Ec(E −1 c (F)) is the largest ec-filter contained in F. Proof. (i). Propositions 2.5 and 2.7 respectively show that Ec(I) is an ec-filter and E−1c (Ec(I)) is an ec-ideal. Now, suppose that K is an ec-ideal containing I. So E−1c (Ec(I)) ⊆ E−1c (Ec(K)) = K. Hence we are done. (ii). This is proved similarly. � The next theorem plays an important role in many of the following results. Theorem 2.12. Let A be a zc-ultrafilter. Then a zero set Z meets every element of Ec(E −1 c (A)) if and only if Z ∈A. Proof. Since A is a filter and Ec(E−1c (A)) ⊆ A, the sufficient condition is evident. For the necessary condition, it is recalled at first that if Z meets every element of A then Z ∈A, see (iv) of Proposition 1.4. Now, we claim that if Z meets every element of Ec(E −1 c (A)) as a particular subfamily of A, then also Z ∈A. Otherwise, for some Z′ ∈A, Z ∩Z′ = ∅. Since the closed sets Z and c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 2 399 A. Veisi Z′ are c-completely separated, there is f ∈ C∗c (X) (in fact 0 ≤ f ≤ 1) such that f(Z) = 1 and f(Z′) = 0. Notice that Z′ ⊆ Z(f) ⊆ Ec� (f), for all �, and; Ec� (f) ∈A, since Z′ ∈A. So Ec� (f) ∈ Ec(E−1c (A)). Now, if � is taken less than 1, then Z ∩Ec� (f) = ∅ which contradicts with our assumption of Z. So Z ∈A and the proof is complete. � The following proposition shows that, as Z−1c (A) is a maximal ideal in Cc(X), E −1 c (A) is also a maximal ideal in C∗c (X), where A is a zc-ultrafilter on X. Proposition 2.13. Let A be a zc-ultrafilter on X. Then: (i) E−1c (A) is a maximal ideal. (ii) E−1c (A) is an ec-ideal. (iii) E−1c (A) = E−1c (Ec(E−1c (A))). Proof. (i). Let M be a maximal ideal of C∗c (X) containing E −1 c (A). Hence Ec(E −1 c (A)) ⊆ Ec(M). Since every element of Ec(M) meets every element of Ec(E −1 c (A)), Theorem 2.12 gives Ec(M) ⊆ A. So M = E−1c (Ec(M)) ⊆ E−1c (A) and hence M = E−1c (A). (ii). It follows by (i). (iii). Since the maximal ideal E−1c (A) is contained in the proper ideal E−1c (Ec(E−1c (A))), the result now holds. � An ec-ultrafilter on X is meant a maximal ec-filter, i.e., one not contained in any other ec-filter. As usual, every ec-filter F is contained in an ec-ultrafilter. This is obtained by considering the collection of all ec-filters containing F and the use of the Zorn’s lemma, where the partially ordered relation on F is inclusion. Proposition 2.14. Let M be an ideal in C∗c (X) and F a zc-filter on X. Then: (i) If M is a maximal ideal then Ec(M) is an ec-ultrafilter. (ii) If F is an ec-ultrafilter then E−1c (F) is a maximal ideal. (iii) If M is an ec-ideal, then M is maximal if and only if Ec(M) is an ec-ultrafilter. (iv) If F is an ec-filter, then F is ec-ultrafilter if and only if E−1c (F) is a maximal ideal. Proof. (i). Note that M = E−1c (Ec(M)). Let F′ be an ec-ultrafilter containing Ec(M), then M ⊆ E−1c (F′) and hence M = E−1c (F′). Therefore Ec(M) = Ec(E −1 c (F′)) = F′, which yields the result. (ii). Let M be a maximal ideal of C∗c (X) containing E −1 c (F). Then F ⊆ Ec(M). Hence F = Ec(M) and so E−1c (F) = M. The proofs of (iii) and (iv) are similarly done and further details are omitted. � Corollary 2.15. There is a one-to-one correspondence ψ between the sets of maximal ideals of C∗c (X) and ec-ultrafilters on X, defined by ψ(M) = Ec(M). Proposition 2.16. Let A be a zc-ultrafilter. Then it is the unique zc-ultrafilter containing Ec(E −1 c (A)), and also Ec(E−1c (A)) is the unique ec-ultrafilter con- tained in A. Hence every ec-ultrafilter is contained in unique zc-ultrafilter. c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 2 400 ec-Filters and ec-ideals in the functionally countable subalgebra of C ∗(X) Proof. Let B be a zc-ultrafilter containing Ec(E−1c (A)) and Z ∈ B. Since Z meets every element of Ec(E −1 c (A)), Theorem 2.12 gives B ⊆ A and hence B = A. So the first part of the proposition holds. Now, let K be an ec- ultrafilter contained in A. Then K = Ec(E−1c (K)) ⊆ Ec(E−1c (A)). Since the latter set is an ec-filter, the inclusion cannot be proper, i.e., K = Ec(E−1c (A)). Hence the result is obtained. � Corollary 2.17. The zc-ultrafilters are in one-to-one correspondence with the ec-ultrafilters. Proof. Consider the mapping ψ from the set of zc-ultrafilters into the set of ec-ultrafilters defined by ψ(A) = Ec(E−1c (A)). If ψ(A) = ψ(B), then we have that Ec(E −1 c (A)) = Ec(E−1c (B)) and it is contained in both A and B. So each element of B meets each element of Ec(E−1c (A)). Now, Theorem 2.12 gives B ⊆ A. Similarly, A ⊆ B. Therefore ψ is one-one. Let K be an ec- ultrafilter and A the unique zc-ultrafilter containing it (Proposition 2.16). Then K = Ec(E−1c (A)) and hence ψ(A) = K. Therefore ψ is onto. � Our next two theorems are applications that are based on the concepts of ec-filters and ec-ideals. In the first result (Theorem 2.18) we show that the maximal ideals of Cc(X) are in one-to-one correspondence with those ones of C∗c (X) and the second result (Theorem 2.20) involves characterization of maximal ideals of C∗c (X). Theorem 2.18. Let M (resp. M∗) be the set of maximal ideals of Cc(X) (resp. C∗c (X)). Then M and M∗ have the same cardinality. Proof. If M ∈M then Zc[M] is a zc-ultrafilter and hence E−1c (Zc[M]) ∈M∗, see Propositions 1.4 and 2.13. Define ϕ : M→M∗ which M 7→ E−1c (Zc[M]). If ϕ(M) = ϕ(M′) then Ec(E −1 c (Zc[M])) = Ec(E −1 c (Zc[M ′])) and it is con- tained in both Zc[M] and Zc[M ′]. Since each element of Zc[M ′] meets each element of Ec(E −1 c (Zc[M])), Theorem 2.12 yields Zc[M ′] ⊆ Zc[M]. Similarly, Zc[M] ⊆ Zc[M′]. Therefore M = M′. This verifies ϕ is one-one. To show that ϕ is onto, suppose that M∗ ∈ M∗. Hence Ec(M∗) is an ec-ultrafilter (Proposition 2.14). Now, let A be the unique zc-ultrafilter containing Ec(M∗) (Proposition 2.16), then Z−1c [A] is a maximal ideal in Cc(X) (Proposition 1.4) and M∗ = E−1c (A). Recall that if F is a zc-filter, then we always have Zc[Z −1 c [F]] = F and Ec(E−1c (F)) ⊆ F, but the equality occurs if F is an ec- filter. Now, if we let M = Z−1c [A] then ϕ(M) = E−1c (Zc[M]) = E−1c (A) = M∗. Hence ϕ is onto, which it completes the proof. � Remark 2.19. Combining Corollaries 1.5, 2.15 and 2.17 gives another proof of the above theorem. Theorem 2.20. Let M be an ideal in C∗c (X). Then M is maximal if and only if whenever f ∈ C∗c (X) and each Ec� (f) meets every element of Ec(M), then f ∈ M. c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 2 401 A. Veisi Proof. Necessity: Suppose that f /∈ M. So (M,f) = C∗c (X). Hence h+fg = 1, for some h ∈ M and g ∈ C∗c (X). Let u be an upper bound for g and 0 < � < 1. Then ∅ = Ec� (1) = E c � (h + fg) ⊇ E c � 2 (h) ∩Ec� 2 (fg) ⊇ Ec� 2 (h) ∩Ec� 2u (f), which contradicts with our assumption, since Ec� 2 (h) ∩Ec� 2u (f) 6= ∅. So we are done. Sufficiency: Let M′ be a maximal ideal of C∗c (X) containing M and f ∈ M′. Then Ec(M) ⊆ Ec(M′) and Ec� (f) ∈ Ec(M′), for all �. Since Ec(M′) is an ec- filter, Ec� (f) meets every element of Ec(M ′). Hence it also meets each element of Ec(M). Now, by hypothesis f ∈ M. Therefore M = M′, which gives the result. � 3. Uniform norm topology on C∗c (X) and related closed ideals Consider the supremum-norm on C∗(X), i.e., ‖f‖ = supx∈X |f(x)|, where f ∈ C∗(X). So its restriction on C∗c (X) is also the supremum-norm. This defines a metric d as usual, d(f,g) = ‖f −g‖. The resulting metric topology is called the uniform norm topology on C∗c (X). Convergence in this topology is uniform convergence of the functions. A base for the neighborhood system at g consists of all sets of the form {f : ‖f −g‖≤ �} (� > 0). Equivalently, a base at g is given by all sets {f : |f(x) −g(x)| ≤ u(x) for every x ∈ X}, where u is a positive unit of C∗c (X). If I is an ideal in C∗c (X) then its closure in C ∗ c (X) is denoted by clI. Proposition 3.1. Let I be an ideal in C∗c (X). Then: (i) clI is ideal. (ii) If I is a proper ideal then clI is also a proper ideal. (iii) If I is an ec-ideal then it is a closed ideal. Proof. (i). Let f,g ∈ clI, h ∈ C∗c (X) and let u be an upper bound for h and � > 0 is fixed. Then for some f′ ∈ N� 2 (f) ∩ I and g′ ∈ N� 2 (g) ∩ I we have f′ + g′ ∈ N�(f + g) ∩ I. Moreover, there exists f1 ∈ N � u (f) ∩ I and hence f1h ∈ N�(fh) ∩I. So clI contains f + g and fh. Hence clI is ideal. (ii). If clI is not a proper ideal then 1 ∈ clI and hence N�(1) ∩I contains a unit element of C∗c (X) such as f, since 1−� < f < 1 + � gives f is bounded away from zero (of course, when 0 < � < 1). But this is impossible since f ∈ I. Thus clI is a proper ideal. (iii). Let g ∈ clI and � > 0 arbitrary. Then for some f ∈ N� 2 (g) ∩ I and all x ∈ Ec� 2 (f), we have |g(x)| = |g(x) −f(x) + f(x)| ≤ |g(x) −f(x)| + |f(x)| ≤ � 2 + � 2 = �. c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 2 402 ec-Filters and ec-ideals in the functionally countable subalgebra of C ∗(X) Hence Ec� 2 (f) ⊆ Ec� (g). Since the zc-filter Ec(I) contains Ec� 2 (f), it also contains Ec� (g), for all �. So g ∈ E−1c (Ec(I)) = I and therefore clI ⊆ I. This proves that I is closed and hence the proof is complete. � Immediately, we find there is no proper dense ideal in C∗c (X), and further maximal ideals of C∗c (X) and hence every intersection of them are closed, see Corollary 2.8 and (iii) of the above proposition. We recall that [6, 1D(1)] plays a useful role in the context of C(X). The following is the counterpart for Cc(X). Proposition 3.2. If f,g ∈ Cc(X) and Z(f) is a neighborhood of Z(g), then f = gh for some h ∈ Cc(X). In the remainder of this section, the zero-dimensional topological space X will be assumed to be compact. Hence it is c-pseudo-compact, i.e, Cc(X) = C∗c (X). Lemma 3.3. Let X be a compact space, I an ideal in Cc(X), f ∈ Cc(X) and Z(f) a neighborhood of ∩Zc[I]. Then f ∈ I. Proof. First, we recall that X is compact if and only if the intersection of members of any collection consisting of nonempty closed subsets of X with the finite intersection property (i.e., the intersection of each of a finite number of them is nonempty) is nonempty. The lemma is obvious when I = Cc(X). Now, if I is a proper ideal in Cc(X) then Zc[I] satisfies the finite intersection property and hence ∩Zc[I] 6= ∅. By assumption ∩Zc[I] ⊆ intZ(f). Hence X rintZ(f) ⊆ ⋃ g∈I coz(g) and so X = ⋃ g∈I coz(g)∪intZ(f). By compactness of X, there are a finite number of elements of I, say g1,g2, . . . ,gn, such that X = n⋃ i=1 coz(gi) ∪ intZ(f). Now, if we let g = ∑n i=1 g 2 i then g ∈ I and ∅ 6= Z(g) = ⋂n i=1 Z(gi) ⊆ intZ(f). In view of Proposition 3.2, f is a multiple of g and hence it is contained in I. So the proof is complete. � Proposition 3.4. If g ∈ Cc(X) and � > 0 is fixed, then there exists f ∈ Cc(X) such that ‖g −f‖≤ � and Z(f) is a neighborhood of Z(g). Proof. The trivial solution is f = g, of course when Z(g) is open. In general, it suffices to define f(x) =   g(x) − �, if x ∈ g−1([�, +∞)) 0, if x ∈ Ec� (g) g(x) + �, if x ∈ g−1((−∞,−�]). We note that X is the union of three closed sets g−1([�, +∞)), Ec� (g) and g−1((−∞,−�]) and further f is continuous on each of them. Therefore f is continuous on X, i.e., f ∈ C(X). Notice that the definition of f makes the c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 2 403 A. Veisi cardinality of the range of f the same cardinality of the range of g. Hence this leads us f ∈ Cc(X). Moreover, ‖g −f‖≤ �. Evidently, Z(g) ⊆ g−1((−�,�)) ⊆ intZ(f) which yields Z(f) is a neighborhood of Z(g). � Theorem 3.5. Let I be a proper ideal in Cc(X), I = ∩{Mcp : Mcp ⊇ I} and J = {g ∈ Cc(X) : Z(g) ⊇∩Zc[I]}. Then: (i) I = J. (ii) ∩Zc[I] = ∩Zc[I]. Proof. (i). Let g ∈ J and Mcp be a fixed maximal ideal of Cc(X) containing I. Then Z(g) ⊇ ∩Zc[I] ⊇ ∩Zc[Mcp] = {p}. So g(p) = 0 and hence g ∈ Mcp. Therefore g ∈ I. For the reverse inclusion, we show that if g /∈ J then g /∈ I. If g /∈ J then there exists x ∈ ∩Zc[I] r Z(g). So I ⊆ Mcx but g /∈ Mcx. This means that g /∈ I. The proof of (i) is now complete. (ii). By (i), we have ∩Zc[I] = ∩Zc[J] ⊇ ∩Zc[I]. On the other hand, I ⊆ I implies Zc[I] ⊆ Zc[I] and therefore ∩Zc[I] ⊇∩Zc[I]. So it gives the result. � Corollary 3.6. Let I be a proper ideal in Cc(X) and I as defined in Theorem 3.5. Then I = clI. Proof. Since maximal ideals are closed, ⋂ I⊆M M is also closed, where M is a maximal ideal in Cc(X). Therefore clI ⊆ ⋂ I⊆M M ⊆ I. Let g ∈ I and N�(g) is a neighborhood of g. By Proposition 3.4, there is f such that Z(f) is a neighborhood of Z(g) and ‖g −f‖≤ �. Hence, by Theorem 3.5, ∩Zc[I] ⊆ Z(g) ⊆ intZ(f) and therefore Lemma 3.3 implies f ∈ I. Now, since f ∈ N�(g) ∩ I, it gives g ∈ clI. So I ⊆ clI and we are done. � We conclude the article with the following results for the proper ideals of Cc(X). Corollary 3.7 is a consequence of Corollary 2.8, Proposition 3.1 (iii) and Corollary 3.6; by the same results, plus Corollary 3.7 we obtain Corollary 3.8; finally Corollary 3.9 is the combination of Corollaries 3.7 and 3.8. Corollary 3.7. An ideal I of Cc(X) is closed in Cc(X) if and only if it is an intersection of maximal ideals of Cc(X). Corollary 3.8. An ideal I of Cc(X) is an ec-ideal if and only if it is closed in Cc(X). Corollary 3.9. An ideal I of Cc(X) is an ec-ideal if and only if it is an intersection of maximal ideals of Cc(X). Acknowledgements. 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