- Email: [email protected]
On a question of Kaplansky
Topology and its Applications 232 (2017) 98–101
Contents lists available at ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
On a question of Kaplansky Ali Taherifar Department of Mathematics, Yasouj University, Yasouj, Iran
a r t i c l e
i n f o
Article history: Received 30 January 2017 Received in revised form 3 May 2017 Accepted 6 October 2017 Available online 10 October 2017 MSC: primary 54C30, 54C40 secondary 54D45
a b s t r a c t Kaplansky [7] proved that CK (X) is the intersection of all free maximal ideals in C(X) in the case of discrete X, and asked whether the equality holds in general. In this paper we prove that CK (X) coincides with the intersection of all free maximal ideals if and only if every open hemicompact z-compact (i.e., every zero-set contained in it is compact) subset of X is relatively compact or equivalently, every open Lindelöf z-compact subset of X is relatively compact. We conclude that the equality holds whenever X is a strongly isocompact space. © 2017 Elsevier B.V. All rights reserved.
Keywords: Compact space Realcompact space Hemicompact space Lindelöf space Relatively pseudocompact Continuous function
1. Introduction The equality of CK (X), the set of functions with compact support, with the intersection of the free maximal ideals was first proved for discrete spaces by Kaplansky [7], who asked if the equality holds in general. The Tychonoff plank example [5, 8.20] shows that the equality CK (X) with the intersection of the free maximal ideals (i.e., M βX\X ) may be fail in general. Kohls [8], Gillman–Jerison [5], and Robinson [10] have proved the result for P-spaces, realcompact spaces, and spaces admitting a complete uniform structure, respectively. Robinson [11] and Mandelker [9] have found some equivalent conditions to this equality. In this paper first we call a subset H of X a z-compact subset if every zero-set of X contained in H is compact. Clearly every relatively compact subset (i.e., a subset with compact closure, see [8]) is z-compact. But we will see an example of a z-compact subset which is not relatively compact. Next, we give a characterization of an open Lindelöf z-compact subset of X and use it to find several new equivalent conditions with the fact that CK (X) coincides with the intersection of all free maximal ideals. This result implies that whenever X is a strongly isocompact space, then CK (X) is equal to the intersection of all free maximal ideals. E-mail addresses: [email protected], [email protected]. https://doi.org/10.1016/j.topol.2017.10.004 0166-8641/© 2017 Elsevier B.V. All rights reserved.
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In this paper X is a completely regular Hausdorff space and C(X) is the ring of continuous real valued ˆ functions. The Stone–Cech compactification of X is denoted by βX. If I is an ideal of C(X), then θ(I) = cl Z(f ). The set of all f ∈ C(X) with pseudocompact closure is denoted by Cψ (X) and coincides f ∈I βX βX\υX with M [6], where υX is the realcompactification of X. In [1], we called a maximal ideal M p of C(X) almost real maximal if Z[M p ] is either fixed or contains a free z-filter which is closed under countable intersection. We denoted by λX the set of all p ∈ βX such that M p is an almost real maximal. We have shown that υX ⊆ λX ⊆ βX and M λX\X is the set of all f ∈ C(X) for which X \ Z(f ) is Lindelöf, see Theorem 3.8 in [1]. The reader is referred to [5] and [3] for undefined terms and notations. 2. When is CK (X) equal to the intersection of all free maximal ideals We call a subset A of X, z-compact if every zero-set of X contained in A is compact. Proposition 5.4 in [1] shows that an open locally compact σ-compact subset is z-compact if and only if C∞ (X) coincides with the intersection of all free maximal ideals or equivalently, every open locally compact σ-compact subset of X is relatively pseudocompact. Trivially every compact subset of X is z-compact. But the next result shows that we may have an open Lindelöf subset of X which is not z-compact. First, we need the following lemma which is a characterization of an open Lindelöf subset of X. Lemma 2.1. An open subset A is Lindelöf if and only if A = X \ Z(f ), for some f ∈ M λX\X . Proof. Let A be an open Lindelöf subset of X. By complete regularity of X, there is a family {fα : α ∈ S} of continuous functions such that A = α∈S (X \Z(fα )). So there is a countable subset {f1 , ....fn , ...} such that 1 A = i∈N (X \ Z(fi )). Hence A = X \ Z(f ), where f = i∈N 2i .|fi |/(1 + |fi |). By the Weierstrass M-test, f ∈ C(X) and f ∈ M λX\X , by Theorem 3.8 in [1]. Conversely, if A = X \ Z(f ), for some f ∈ M λX\X , then (X \ Z(f )) is Lindelöf, by Theorem 3.8 in [1]. So we are done. 2 Proposition 2.2. The following statements are equivalent (1) Every open Lindelöf subset of X is z-compact. (2) M βX\X = M λX\X . (3) λX \ X is dense in βX \ X. Proof. (1)⇒(2) Since λX \ X ⊆ βX \ X, M βX\X ⊆ M λX\X . Let f ∈ M λX\X . Then by Theorem 3.8 in [1], X \ Z(f ) is an open Lindelöf subset of X. So by hypothesis, this is a z-compact subset and hence for each g ∈ C(X), Z(1 − f g) ⊆ X \ Z(f ) is compact, i.e., f ∈ M βX\X , by Theorem 2.2 in [4]. (2)⇒(1) Suppose A be an open Lindelöf subset of X. Then by Lemma 2.1, A = X \ Z(f ) for some f ∈ M λX\X . Let Z(g) ⊆ A be a zero-set. Then h = f /(f 2 + g 2 ) ∈ C(X) and Z(g) ⊆ Z(1 − hf ). Now by hypothesis and Theorem 2.2 in [4], Z(g) is compact. (2)⇒(3) The hypothesis implies that clβX (λX \ X) = θ(M λX\X ) = θ(M βX\X ) = clβX (βX \ X) and hence λX \ X is dense in βX \ X. (3)⇒(2) Assume that λX \ X is dense in βX \ X. Then clβX (λX \ X) ∩ (βX \ X) = βX \ X. Therefore βX \ X ⊆ clβX (λX \ X). This implies that M λX\X ⊆ M βX\X . Always we have M βX\X ⊆ M λX\X . Hence we are done. 2 Whenever X is pseudocompact we have βX = λX = υX and hence M βX\X = M λX\X , so by the above proposition every open Lindelöf subset of X is z-compact. Remark 2.3. If A ⊆ X is relatively compact, then A is z-compact. But the converse fails. For, consider the Tychonoff plank T in [5, 8.20]. Then T is pseudocompact and CK (T ) is not equal with M βX\X = M λX\X .
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Hence there is f ∈ M λX\X such that clX (X \ Z(f )) is not compact, i.e., X \ Z(f ) is not relatively compact. On the other hand X \ Z(f ) is an open Lindelöf set, and hence is z-compact by Proposition 2.2. In the next result we give a characterization of an open hemicompact z-compact subset. Lemma 2.4. Let A be an open subset of X. The following are equivalent. (1) (2) (3) (4)
A is hemicompact and z-compact. A is σ-compact and z-compact. A = X \ Z(f ) for some f ∈ M βX\X . A is Lindelöf and z-compact.
Proof. (1)⇒(2) See [12, 17I1]. ∞ (2)⇒(3) A is σ-compact, so there are compact subsets A1 , A2 , ... such that A = i=1 Ai . For each i, we have (X \ A) ∩ Ai = ∅. Thus they are completely separated (since X is a completely regular space), i.e., for each i ∈ N, there is an fi ∈ C(X) such that fi (X) ⊆ [0, 1], fi (Ai ) = 1 and fi (X \ A) = 0. Now we define ∞ f = i=1 fi /2i . By the Weierstrass M-test, f ∈ C(X) and trivially A = X \ Z(f ). On the other hand for each g ∈ C(X), we have Z(1 − f g) ⊆ X \ Z(f ) = A. So by hypothesis, Z(1 − f g) is compact for each g ∈ C(X) and hence f ∈ M βX\X , by [4, Theorem 2.2]. ∞ (3)⇒(4) We have A = n=1 {x ∈ X : |f (x)| ≥ 1/n}. For each n ∈ N, the set {x ∈ X : |f (x)| ≥ 1 |f | ≥ n1 f 1/n} = Z(1 − f φn (f )), where φn (f ) = , see Lemma 1.2 in [1]. Hence for each n ∈ N, n2 f |f | ≤ n1 {x ∈ X : |f (x)| ≥ 1/n} is compact, by [4, Theorem 2.2]. Thus A is Lindelöf. Now let Z(g) ⊆ (X \ Z(f )) be 2 +1 2 a zero-set. Then Z(f ) ∩ Z(g) = ∅. So the function h = fg2 +g 2 is continuous and Z(g) ⊆ Z(1 − hf ). But by [4, Theorem 2.2], Z(1 − hf 2 ) is compact. Thus Z(g) is compact, i.e., A = X \ Z(f ) is z-compact. (4)⇒(1) It is enough to show that A is hemicompact. By complete regularity of X, there is a family {fα : α ∈ S} of continuous functions such that A = α∈S (X \ Z(fα )). Hence there exists a countable subset H of S such that A = α∈H (X \ Z(fα )). But for each α ∈ H, there is a countable family {Z(gαn ) : n ∈ N} ∞ ∞ of zero-sets such that X \ Z(fα ) = n=1 Z(gαn ) = n=1 intX Z(gαn ), where for each n ∈ N, Z(gαn ) = {x ∈ X : |fα (x)| ≥ 1/n} and by z-compactness of A each Z(gαn ) is compact. Now we claim that the set of all finite unions of elements of the family S = {Z(gαn ) : α ∈ H, n ∈ N} is the set of compact subsets of A which we need. To see this, suppose that K be a compact subset of A. Then there are a finite subfamily n n {Z1 , ..., Zn } of S such that K ⊆ i=1 intX Zi ⊆ i=1 Zi . This shows that A is a hemicompact subset. 2 It is well known that every hemicompact space is σ-compact and a σ-compact space is a Lindelöf space. But the converse fails. However from the above result we get that an open z-compact subset is hemicompact if and only if it is σ-compact if and only if it is Lindelöf. Now we are ready to present our main result of this paper. Theorem 2.5. The following statements are equivalent. (1) (2) (3) (4) (5)
The ideal CK (X) coincides with the intersection of all free maximal ideals. Every open hemicompact z-compact subset of X is relatively compact. Every open σ-compact z-compact subset of X is relatively compact. Every open Lindelöf z-compact subset of X is relatively compact. Every relatively pseudocompact z-compact cozero-set is relatively compact.
Proof. Lemma 2.4 implies that (1), (2), (3) and (4) are equivalent.
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(4)⇒(5) Let A = X \ Z(f ) be a relatively pseudocompact z-compact cozero-set. Then the z-compactness of A implies that A is Lindelöf and so by hypothesis, A is relatively compact. (5)⇒(4) Now we assume that A is an open Lindelöf z-compact subset of X. Then by Lemma 2.4, A = X \ Z(f ), for some f ∈ M βX\X . Thus f ∈ M βX\υX , i.e., A is relatively pseudocompact (since clX (X \ Z(f )) is pseudocompact), and hence by hypothesis, A is relatively compact. 2 Recall from [2] that a space X is strongly isocompact if X is isocompact (i.e., every closed countably compact subspace is compact) and has weak property D (i.e., every infinite closed discrete subset of X has an infinite subset that is C-embedded in X). The class of strongly isocompact spaces include the class of topologically complete spaces (hence all realcompact and paracompact spaces) and the class of normal isocompact spaces (e.g., metric spaces), see [2, Corollary 3.2]. Corollary 2.6. If X is a strongly isocompact space, then CK (X) is the intersection of all free maximal ideals. Proof. This follows from [2, 3.0(2)] and Theorem 2.5.
2
Acknowledgement The author would like to thank the referee for the careful reading of this paper. References [1] F. Azarpanah, M. Ghirati, A. Taherifar, Closed ideals in C(X) with different representations, Houst. J. Math. (2016), in press. [2] R.L. Blair, Eric K. van Douwen, Nearly realcompact spaces, Topol. Appl. 47 (1992) 209–221. [3] R. Engelking, General Topology, Sigma Ser. Pure Math., vol. 6, Heldermann Verlag, Berlin, 1989. [4] M. Ghirati, A. Taherifar, Intersections of essential (resp., free) maximal ideals of C(X), Topol. Appl. 167 (2014) 62–68. [5] L. Gillman, M. Jerison, Rings of Continuous Functions, Springer, 1976. [6] D. Johnson, M. Mandelker, Functions with pseudocompact supports, Gen. Topol. Appl. 3 (1973) 331–338. [7] L. Kaplansky, Topological rings, Am. J. Math. 69 (1947) 153–183. [8] C.W. Kohls, Ideals in rings of continuous functions, Fundam. Math. 45 (1975) 28–50. [9] M. Mandelker, Supports of continuous functions, Trans. Am. Math. Soc. 156 (1971) 73–83. [10] S.M. Robinson, The intersection of the free maximal ideals in a complete space, Proc. Am. Math. Soc. 17 (1966) 468–469. [11] S.M. Robinson, A note on the intersection of free maximal ideals, J. Aust. Math. Soc. 10 (1969) 41–54. [12] S. Willard, General Topology, Addison-Wesley Publishing Company, Inc., 1970.
Contents lists available at ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
On a question of Kaplansky Ali Taherifar Department of Mathematics, Yasouj University, Yasouj, Iran
a r t i c l e
i n f o
Article history: Received 30 January 2017 Received in revised form 3 May 2017 Accepted 6 October 2017 Available online 10 October 2017 MSC: primary 54C30, 54C40 secondary 54D45
a b s t r a c t Kaplansky [7] proved that CK (X) is the intersection of all free maximal ideals in C(X) in the case of discrete X, and asked whether the equality holds in general. In this paper we prove that CK (X) coincides with the intersection of all free maximal ideals if and only if every open hemicompact z-compact (i.e., every zero-set contained in it is compact) subset of X is relatively compact or equivalently, every open Lindelöf z-compact subset of X is relatively compact. We conclude that the equality holds whenever X is a strongly isocompact space. © 2017 Elsevier B.V. All rights reserved.
Keywords: Compact space Realcompact space Hemicompact space Lindelöf space Relatively pseudocompact Continuous function
1. Introduction The equality of CK (X), the set of functions with compact support, with the intersection of the free maximal ideals was first proved for discrete spaces by Kaplansky [7], who asked if the equality holds in general. The Tychonoff plank example [5, 8.20] shows that the equality CK (X) with the intersection of the free maximal ideals (i.e., M βX\X ) may be fail in general. Kohls [8], Gillman–Jerison [5], and Robinson [10] have proved the result for P-spaces, realcompact spaces, and spaces admitting a complete uniform structure, respectively. Robinson [11] and Mandelker [9] have found some equivalent conditions to this equality. In this paper first we call a subset H of X a z-compact subset if every zero-set of X contained in H is compact. Clearly every relatively compact subset (i.e., a subset with compact closure, see [8]) is z-compact. But we will see an example of a z-compact subset which is not relatively compact. Next, we give a characterization of an open Lindelöf z-compact subset of X and use it to find several new equivalent conditions with the fact that CK (X) coincides with the intersection of all free maximal ideals. This result implies that whenever X is a strongly isocompact space, then CK (X) is equal to the intersection of all free maximal ideals. E-mail addresses: [email protected], [email protected]. https://doi.org/10.1016/j.topol.2017.10.004 0166-8641/© 2017 Elsevier B.V. All rights reserved.
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In this paper X is a completely regular Hausdorff space and C(X) is the ring of continuous real valued ˆ functions. The Stone–Cech compactification of X is denoted by βX. If I is an ideal of C(X), then θ(I) = cl Z(f ). The set of all f ∈ C(X) with pseudocompact closure is denoted by Cψ (X) and coincides f ∈I βX βX\υX with M [6], where υX is the realcompactification of X. In [1], we called a maximal ideal M p of C(X) almost real maximal if Z[M p ] is either fixed or contains a free z-filter which is closed under countable intersection. We denoted by λX the set of all p ∈ βX such that M p is an almost real maximal. We have shown that υX ⊆ λX ⊆ βX and M λX\X is the set of all f ∈ C(X) for which X \ Z(f ) is Lindelöf, see Theorem 3.8 in [1]. The reader is referred to [5] and [3] for undefined terms and notations. 2. When is CK (X) equal to the intersection of all free maximal ideals We call a subset A of X, z-compact if every zero-set of X contained in A is compact. Proposition 5.4 in [1] shows that an open locally compact σ-compact subset is z-compact if and only if C∞ (X) coincides with the intersection of all free maximal ideals or equivalently, every open locally compact σ-compact subset of X is relatively pseudocompact. Trivially every compact subset of X is z-compact. But the next result shows that we may have an open Lindelöf subset of X which is not z-compact. First, we need the following lemma which is a characterization of an open Lindelöf subset of X. Lemma 2.1. An open subset A is Lindelöf if and only if A = X \ Z(f ), for some f ∈ M λX\X . Proof. Let A be an open Lindelöf subset of X. By complete regularity of X, there is a family {fα : α ∈ S} of continuous functions such that A = α∈S (X \Z(fα )). So there is a countable subset {f1 , ....fn , ...} such that 1 A = i∈N (X \ Z(fi )). Hence A = X \ Z(f ), where f = i∈N 2i .|fi |/(1 + |fi |). By the Weierstrass M-test, f ∈ C(X) and f ∈ M λX\X , by Theorem 3.8 in [1]. Conversely, if A = X \ Z(f ), for some f ∈ M λX\X , then (X \ Z(f )) is Lindelöf, by Theorem 3.8 in [1]. So we are done. 2 Proposition 2.2. The following statements are equivalent (1) Every open Lindelöf subset of X is z-compact. (2) M βX\X = M λX\X . (3) λX \ X is dense in βX \ X. Proof. (1)⇒(2) Since λX \ X ⊆ βX \ X, M βX\X ⊆ M λX\X . Let f ∈ M λX\X . Then by Theorem 3.8 in [1], X \ Z(f ) is an open Lindelöf subset of X. So by hypothesis, this is a z-compact subset and hence for each g ∈ C(X), Z(1 − f g) ⊆ X \ Z(f ) is compact, i.e., f ∈ M βX\X , by Theorem 2.2 in [4]. (2)⇒(1) Suppose A be an open Lindelöf subset of X. Then by Lemma 2.1, A = X \ Z(f ) for some f ∈ M λX\X . Let Z(g) ⊆ A be a zero-set. Then h = f /(f 2 + g 2 ) ∈ C(X) and Z(g) ⊆ Z(1 − hf ). Now by hypothesis and Theorem 2.2 in [4], Z(g) is compact. (2)⇒(3) The hypothesis implies that clβX (λX \ X) = θ(M λX\X ) = θ(M βX\X ) = clβX (βX \ X) and hence λX \ X is dense in βX \ X. (3)⇒(2) Assume that λX \ X is dense in βX \ X. Then clβX (λX \ X) ∩ (βX \ X) = βX \ X. Therefore βX \ X ⊆ clβX (λX \ X). This implies that M λX\X ⊆ M βX\X . Always we have M βX\X ⊆ M λX\X . Hence we are done. 2 Whenever X is pseudocompact we have βX = λX = υX and hence M βX\X = M λX\X , so by the above proposition every open Lindelöf subset of X is z-compact. Remark 2.3. If A ⊆ X is relatively compact, then A is z-compact. But the converse fails. For, consider the Tychonoff plank T in [5, 8.20]. Then T is pseudocompact and CK (T ) is not equal with M βX\X = M λX\X .
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Hence there is f ∈ M λX\X such that clX (X \ Z(f )) is not compact, i.e., X \ Z(f ) is not relatively compact. On the other hand X \ Z(f ) is an open Lindelöf set, and hence is z-compact by Proposition 2.2. In the next result we give a characterization of an open hemicompact z-compact subset. Lemma 2.4. Let A be an open subset of X. The following are equivalent. (1) (2) (3) (4)
A is hemicompact and z-compact. A is σ-compact and z-compact. A = X \ Z(f ) for some f ∈ M βX\X . A is Lindelöf and z-compact.
Proof. (1)⇒(2) See [12, 17I1]. ∞ (2)⇒(3) A is σ-compact, so there are compact subsets A1 , A2 , ... such that A = i=1 Ai . For each i, we have (X \ A) ∩ Ai = ∅. Thus they are completely separated (since X is a completely regular space), i.e., for each i ∈ N, there is an fi ∈ C(X) such that fi (X) ⊆ [0, 1], fi (Ai ) = 1 and fi (X \ A) = 0. Now we define ∞ f = i=1 fi /2i . By the Weierstrass M-test, f ∈ C(X) and trivially A = X \ Z(f ). On the other hand for each g ∈ C(X), we have Z(1 − f g) ⊆ X \ Z(f ) = A. So by hypothesis, Z(1 − f g) is compact for each g ∈ C(X) and hence f ∈ M βX\X , by [4, Theorem 2.2]. ∞ (3)⇒(4) We have A = n=1 {x ∈ X : |f (x)| ≥ 1/n}. For each n ∈ N, the set {x ∈ X : |f (x)| ≥ 1 |f | ≥ n1 f 1/n} = Z(1 − f φn (f )), where φn (f ) = , see Lemma 1.2 in [1]. Hence for each n ∈ N, n2 f |f | ≤ n1 {x ∈ X : |f (x)| ≥ 1/n} is compact, by [4, Theorem 2.2]. Thus A is Lindelöf. Now let Z(g) ⊆ (X \ Z(f )) be 2 +1 2 a zero-set. Then Z(f ) ∩ Z(g) = ∅. So the function h = fg2 +g 2 is continuous and Z(g) ⊆ Z(1 − hf ). But by [4, Theorem 2.2], Z(1 − hf 2 ) is compact. Thus Z(g) is compact, i.e., A = X \ Z(f ) is z-compact. (4)⇒(1) It is enough to show that A is hemicompact. By complete regularity of X, there is a family {fα : α ∈ S} of continuous functions such that A = α∈S (X \ Z(fα )). Hence there exists a countable subset H of S such that A = α∈H (X \ Z(fα )). But for each α ∈ H, there is a countable family {Z(gαn ) : n ∈ N} ∞ ∞ of zero-sets such that X \ Z(fα ) = n=1 Z(gαn ) = n=1 intX Z(gαn ), where for each n ∈ N, Z(gαn ) = {x ∈ X : |fα (x)| ≥ 1/n} and by z-compactness of A each Z(gαn ) is compact. Now we claim that the set of all finite unions of elements of the family S = {Z(gαn ) : α ∈ H, n ∈ N} is the set of compact subsets of A which we need. To see this, suppose that K be a compact subset of A. Then there are a finite subfamily n n {Z1 , ..., Zn } of S such that K ⊆ i=1 intX Zi ⊆ i=1 Zi . This shows that A is a hemicompact subset. 2 It is well known that every hemicompact space is σ-compact and a σ-compact space is a Lindelöf space. But the converse fails. However from the above result we get that an open z-compact subset is hemicompact if and only if it is σ-compact if and only if it is Lindelöf. Now we are ready to present our main result of this paper. Theorem 2.5. The following statements are equivalent. (1) (2) (3) (4) (5)
The ideal CK (X) coincides with the intersection of all free maximal ideals. Every open hemicompact z-compact subset of X is relatively compact. Every open σ-compact z-compact subset of X is relatively compact. Every open Lindelöf z-compact subset of X is relatively compact. Every relatively pseudocompact z-compact cozero-set is relatively compact.
Proof. Lemma 2.4 implies that (1), (2), (3) and (4) are equivalent.
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(4)⇒(5) Let A = X \ Z(f ) be a relatively pseudocompact z-compact cozero-set. Then the z-compactness of A implies that A is Lindelöf and so by hypothesis, A is relatively compact. (5)⇒(4) Now we assume that A is an open Lindelöf z-compact subset of X. Then by Lemma 2.4, A = X \ Z(f ), for some f ∈ M βX\X . Thus f ∈ M βX\υX , i.e., A is relatively pseudocompact (since clX (X \ Z(f )) is pseudocompact), and hence by hypothesis, A is relatively compact. 2 Recall from [2] that a space X is strongly isocompact if X is isocompact (i.e., every closed countably compact subspace is compact) and has weak property D (i.e., every infinite closed discrete subset of X has an infinite subset that is C-embedded in X). The class of strongly isocompact spaces include the class of topologically complete spaces (hence all realcompact and paracompact spaces) and the class of normal isocompact spaces (e.g., metric spaces), see [2, Corollary 3.2]. Corollary 2.6. If X is a strongly isocompact space, then CK (X) is the intersection of all free maximal ideals. Proof. This follows from [2, 3.0(2)] and Theorem 2.5.
2
Acknowledgement The author would like to thank the referee for the careful reading of this paper. References [1] F. Azarpanah, M. Ghirati, A. Taherifar, Closed ideals in C(X) with different representations, Houst. J. Math. (2016), in press. [2] R.L. Blair, Eric K. van Douwen, Nearly realcompact spaces, Topol. Appl. 47 (1992) 209–221. [3] R. Engelking, General Topology, Sigma Ser. Pure Math., vol. 6, Heldermann Verlag, Berlin, 1989. [4] M. Ghirati, A. Taherifar, Intersections of essential (resp., free) maximal ideals of C(X), Topol. Appl. 167 (2014) 62–68. [5] L. Gillman, M. Jerison, Rings of Continuous Functions, Springer, 1976. [6] D. Johnson, M. Mandelker, Functions with pseudocompact supports, Gen. Topol. Appl. 3 (1973) 331–338. [7] L. Kaplansky, Topological rings, Am. J. Math. 69 (1947) 153–183. [8] C.W. Kohls, Ideals in rings of continuous functions, Fundam. Math. 45 (1975) 28–50. [9] M. Mandelker, Supports of continuous functions, Trans. Am. Math. Soc. 156 (1971) 73–83. [10] S.M. Robinson, The intersection of the free maximal ideals in a complete space, Proc. Am. Math. Soc. 17 (1966) 468–469. [11] S.M. Robinson, A note on the intersection of free maximal ideals, J. Aust. Math. Soc. 10 (1969) 41–54. [12] S. Willard, General Topology, Addison-Wesley Publishing Company, Inc., 1970.