@
Appl. Gen. Topol. 23, no. 1 (2022), 79-90

doi:10.4995/agt.2022.15844

© AGT, UPV, 2022

Closed 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

In this paper, closed ideals in Cc(X), the functionally countable subal-
gebra of C(X), with the mc-topology are studied. We show that if X
is a CUC-space, then C∗c (X) with the uniform norm-topology is a Ba-
nach algebra. Closed ideals in Cc(X) as a modified countable analogue
of closed ideals in C(X) with the m-topology, are characterized. For
a zero-dimensional space X, we show that a proper ideal in Cc(X) is
closed if and only if it is an intersection of maximal ideals of Cc(X). It
is also shown that every ideal in Cc(X) with the mc-topology is closed
if and only if X is a P -space if and only if every ideal in C(X) with the
m-topology is closed. Also, for a strongly zero-dimensional space X, it
is proved that every properly closed ideal in C∗c (X) is an intersection of
maximal ideals of C∗c (X) if and only if X is pseudocompact if and only
if every properly closed ideal in C∗(X) is an intersection of maximal
ideals of C∗(X). Finally, we show that if X is a P -space, then the
family of ec-ultrafilters and zc-ultrafilter coincide.

2020 MSC: 54C30; 54C40; 13C11.

Keywords: zero-dimensional space; functionally countable subalgebra; m-
topology; closed ideal; ec-filter; ec-ideal; P -space.

1. Introduction

In what follows X stands for an infinite completely regular Hausdorff topo-
logical space (i.e., infinite Tychonoff space) and C(X) as usual denotes the
ring of all real-valued continuous functions on X. C∗(X) designates the sub-
ring of C(X) containing all those members which are bounded over X. For

Received 30 June 2021 – Accepted 09 October 2021

http://dx.doi.org/10.4995/agt.2022.15844


A. Veisi

each f ∈ C(X), the zero-set of f, denoted by Z(f), is the set of zeros of f
and X \Z(f) is the cozero-set of f and the set of all zero-sets in X is denoted
by Z(X). An ideal I in C(X) is called a z-ideal if f ∈ I, g ∈ C(X) and
Z(f) ⊆ Z(g), then g ∈ I. The space βX is the Stone-C̆ech compactification of
X and for any p ∈ βX, the maximal ideal Mp of C(X) is the set of all f ∈ C(X)
for which p ∈ clβXZ(f). Moreover, Mp is fixed if and only if p ∈ X (in which
case, we put Mp = Mp = {f ∈ C(X) : p ∈ Z(f)}). Whenever

C(X)
Mp

∼= R,
then Mp is called real, else hyper-real, see [5, Chapter 8]. We recall that a
zero-dimensional space is a Hausdorff space with a base consisting of clopen
(closed-open) sets. A Tychonoff space X is called strongly zero-dimensional if
for every finite cover {Ui}ki=1 of X by cozero-sets there exists a finite refine-
ment {Vi}mi=1 of mutually disjoint open sets. A Tychonoff space X is strongly
zero-dimensional if and only if βX is zero-dimensional, see [2].
The subring of C(X) consisting of those functions with countable (resp. fi-
nite) image, which is denoted by Cc(X) (resp. C

F (X)) is an R-subalgebra of
C(X). The subring C∗c (X) of Cc(X) consists of bounded elements of Cc(X).
So C∗c (X) = C

∗(X) ∩ Cc(X). The rings Cc(X) and CF (X) are introduced
and investigated in [3] and more studied in [1], [4], [9], [10] and [12]. A topo-
logical space X is called countably pseudocompact, briefly, c-pseudocompact if
Cc(X) = C

∗
c (X). A nonempty subfamily F of Zc(X) := {Z(f) : f ∈ Cc(X)}

is called a zc-filter if it is a filter on X. For an ideal I in Cc(X) and a zc-
filter F, we define Zc[I] = {Z(f) : f ∈ I}, ∩Zc[I] = ∩{Z(f) : f ∈ I} and
Z−1c [F] = {f ∈ Cc(X) : Z(f) ∈F}. It is observed that F = Zc[Z−1c [F]]. Also,
Zc[I] is a zc-filter on X and Z

−1
c [Zc[I]] ⊇ I. If the equality holds, then I is

called a zc-ideal. This means that if f ∈ I, g ∈ Cc(X) and Z(f) ⊆ Z(g), then
g ∈ I. So maximal ideals in Cc(X) are zc-ideals. In the same way, for an ideal
I of C∗c (X) and a zc-filter F on X, Ec(I) is an ec-filter and E−1c (F) is an ec-
ideal. The counterpart notions are E−1c (Ec(I)) ⊇ I and Ec(E−1c (F)) = F,
see [14]. By β0X, we mean the Banaschewski compactification of a zero-
dimensional space X. If βX is zero-dimensional, then βX = β0X, see [13,
Section 4.7] for more details. According to [1, Theorems 4.2, 4.8], for any
p ∈ β0X, the maximal ideal Mpc of Cc(X) is the set of all f ∈ Cc(X) for
which p ∈ clβ0XZ(f), or equivalently, it is the set of all f ∈ Cc(X) for which
πp ∈ clβXZ(f). Moreover, Mpc is fixed if and only if p ∈ X (in which case, we
put Mpc = Mcp = {f ∈ Cc(X) : p ∈ Z(f)}). Let S be a subring of C(X) and
a topological space. An ideal I of S is called a closed ideal if I = clSI, briefly,
I = clI. The paper is organized as follows. In Section 2, we introduce the
mc-topology on Cc(X) and derive some corollaries on the ideals of Cc(X) and
C∗c (X). We show that if X is a CUC-space, then C

∗
c (X) with the uniform-norm

topology is a Banach algebra. It is shown that an ideal in Cc(X) is a z-ideal
if and only if it is a zc-ideal. In [5], closed ideals in C(X) with the m-topology
are characterized. In Section 3, the countable analogue of this characterization
is given. We show that a proper ideal in Cc(X) is closed if and only if it is
an intersection of maximal ideals in Cc(X). It is also shown that every ideal

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 80



Closed ideals in the functionally countable subalgebra of C(X)

in Cc(X) is closed if and only if X is a P-space if and only if every ideal in
C(X) is closed. For a strongly zero-dimensional space X, we prove that every
properly closed ideal in C∗c (X) is an intersection of maximal ideals of C

∗
c (X)

if and only if X is pseudocompact if and only if every properly closed ideal in
C∗(X) is an intersection of maximal ideals of C∗(X). Finally, we show that if
X is a P-space, then the family of ec-ultrafilters and zc-ultrafilter coincide.

2. Some properties of ideals in Cc(X)

The m-topology on C(X) was first introduced and studied by Hewitt [8], the
generalizing work of E. H. Moore. In his article, he demonstrated that certain
classes of topological spaces X can be characterized by topological properties of
C(X) with the m-topology. For example, he showed that X is pseudocompact
if and only if C(X) with the m-topology is first countable. Several authors
have investigated the topological properties of X via properties of C(X), for
more information, one can refer to [6] and [11]. The m-topology on C(X) is
defined by taking the sets of the form

B(f,u) = {g ∈ C(X) : |f(x) −g(x)| < u(x) for all x ∈ X},
as a base for the neighborhood system at f, for each f ∈ C(X) and each
positive unit u of C(X). The mc-topology (in brief, mc) on Cc(X) is determined
by considering the sets of the form

B(f,u) = {g ∈ Cc(X) : |f(x) −g(x)| < u(x) for all x ∈ X},
as a base for the neighborhood system at f, for each f ∈ Cc(X) and each
positive unit u of Cc(X). The uniform topology, or the uc-topology (in brief,
uc) on Cc(X) is defined by taking the sets of the form

B(f,ε) = {g ∈ Cc(X) : |f(x) −g(x)| < ε for all x ∈ X},
as a base for the neighborhood system at f, for each f ∈ Cc(X) and each ε > 0.
Equivalently, a base at f is given by all sets

B(f,u) = {g ∈ Cc(X) : |f(x) −g(x)| < u(x) for all x ∈ X},
where u is a positive unit of C∗c (X). We observe that uc ⊆ mc. It is shown
in [15] that uc = mc if and only if X is countably pseudocompact. The uc-
topology turns Cc(X) into a metric space with d(f,g) = ‖f−g‖ = sup{|f(x)−
g(x)| : x ∈ X}. Also, the mc-topology is contained in the relative m-topology.
We remind a well-known result that due to Rudin, Pelczynski and Semadeni
which asserts that a compact Hausdorff space X is functionally countable (i.e.,
C(X) = Cc(X)) if and only if X is scattered. So if X is a compact scattered
space or a countable space, then C(X) = Cc(X), and thus the mc-topology
and the m-topology coincide.

Proposition 2.1. Let I be an ideal in Cc(X) (resp. C
∗
c (X)) and the topology

on Cc(X) be the mc-topology. Then:

(i) cl I is an ideal in Cc(X) (resp. C
∗
c (X)) and hence I is contained in a

closed ideal.
(ii) If I is a proper ideal, then cl I is also a proper ideal and hence there is

no proper dense ideal in Cc(X) (resp. C
∗
c (X)).

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 81



A. Veisi

Proof. We provide the proof for which case I is an ideal in Cc(X). In the same
way, the proof holds for the ideal I in C∗c (X). (i). Clearly, the result holds if
I = Cc(X). Suppose that I $ Cc(X). Let f,g ∈ clI, h ∈ Cc(X) and u be a
positive unit of Cc(X). Then for some f

′ ∈ B(f, u
2

) ∩ I, and g′ ∈ B(g, u
2

) ∩ I,
we have f′ + g′ ∈ B(f + g,u) ∩ I. To show that fh ∈ clI, we consider the
positive unit

u1 =
u

(|h| + 1)(u + 1)
∈ Cc(X).

Therefore, for some f1 ∈ B(f,u1) ∩ I we have that |fh−f1h| < u1|h| < u. So
f1h ∈ B(fh,u)∩I. Moreover, if f ∈ clI, then also −f ∈ clI. Thus, clI contains
both f + g and fh. So clI is ideal. (ii). Suppose that I is a proper ideal in
Cc(X) and clI = Cc(X). Consider the constant function 1 ∈ clI and 0 < ε < 1.
Hence, the nonempty set B(1,ε) ∩ I contains a nonzero element of Cc(X), f
say. Since 1 −ε < f(x) < 1 + ε for each x ∈ X, we have Z(f) = ∅, i.e., f is a
unit of Cc(X), which is impossible (because f ∈ I). Thus, clI $ Cc(X), and
we are done. �

The next result is now immediate.

Corollary 2.2. Any maximal ideal in Cc(X) (resp. C
∗
c (X)) and hence any

intersection of maximal ideals in Cc(X) (resp. C
∗
c (X)) is closed.

Definition 2.3. An ideal I in a commutative ring with unity R is called a
z-ideal in R if for each a ∈ I, we have Ma ⊆ I, here Ma is the intersection of
all maximal ideals in R containing a.

Evidently, each maximal ideal in R is a z-ideal. This notion of z-ideal is
consistent with the notion of z-ideals in C(X), see [5, 4A(5)].

Proposition 2.4. Let X be zero-dimensional and I be an ideal in C∗c (X).
Then I is a z-ideal if and only if g ∈ I whenever Z(fβ) ⊆ Z(gβ) with f ∈ I
and g ∈ C∗c (X), where fβ is the extension of f to βX.

Proof. (⇒) : Let f ∈ I, g ∈ C∗c (X) and Z(fβ) ⊆ Z(gβ) and let Mf be the
intersection of all the maximal ideals in C∗c (X) containing f. By the assump-
tion, Mf ⊆ I. Let M be a maximal ideal in C∗c (X) containing f. According
to [9, Corollary 2.11], M has a form of M∗pc = {h ∈ C∗c (X) : hβ(p) = 0}, for
some p ∈ βX. Now, Z(fβ) ⊆ Z(gβ) implies that g ∈ M. Hence, g ∈ I.
(⇐) : Let f ∈ I and g ∈ Mf . Then f ∈ M∗pc implies that g ∈ M∗pc , i.e.,
Z(fβ) ⊆ Z(gβ). Therefore, by the hypothesis, g ∈ I. �

Lemma 2.5. Let X be zero-dimensional and I be an ideal in Cc(X). Then I
is a z-ideal if and only if it is a zc-ideal.

Proof. (⇒) : Let I be a z-ideal in Cc(X), f ∈ I and Z(f) ⊆ Z(g) with
g ∈ Cc(X). We have to show that g ∈ I. Since I is a z-ideal, we have Mf ⊆ I,
where Mf is the intersection of all the maximal ideals in Cc(X) containing f.
It suffices to show that g ∈ Mf . So let Mpc (p ∈ β0X) be any maximal ideal
in Cc(X) which contains f, we have to show that g ∈ Mpc (see [1, Theorem

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 82



Closed ideals in the functionally countable subalgebra of C(X)

4.2]). Indeed f ∈ Mpc implies that p ∈ clβ0XZ(f) which further implies that
p ∈ clβ0XZ(g), by the assumption, Z(f) ⊆ Z(g). Hence, g ∈ Mpc . Thus, I
becomes a zc-ideal in Cc(X).
(⇐) : Let I be a zc-ideal in Cc(X) and f ∈ I. We must show Mf ⊆ I.
Let g ∈ Mf . Then f ∈ Mpc gives g ∈ Mpc , where p ∈ β0X. Equivalently,
clβ0XZ(f) ⊆ clβ0XZ(g). So Z(f) = clβ0XZ(f) ∩X ⊆ clβ0XZ(g) ∩X = Z(g).
Now, the assumption yields that g ∈ I. �

Proposition 2.6. If I is a closed ideal in Cc(X), then I is a zc-ideal.

Proof. Suppose that Z(f) ⊆ Z(g), f ∈ I and g ∈ Cc(X). To show that g ∈ I,
we show that g ∈ clI because I = clI. Let u ∈ Cc(X) be a positive unit and
let us define a function h : X → R as follows:

h(x) =




g(x)−u(x)
2

f(x)
where g(x) ≥ u(x)

2
,

0 where |g(x)| ≤ u(x)
2
,

g(x)+
u(x)

2

f(x)
where g(x) ≤−u(x)

2
.

From the continuity of h on the three closed sets (g − u
2

)−1([0,∞)), (g +
u
2

)−1([0,∞)) ∩(g− u
2

)−1((−∞, 0]), and (g + u
2

)−1((−∞, 0]), which their union
is X, we infer that h ∈ C(X). Moreover, since the ranges of g,u and f are
countable, the range of h is also countable, i.e., h ∈ Cc(X). Thus, fh ∈ I.
Furthermore, it is easy to see that |g(x) −f(x)h(x)| < u(x) for every x ∈ X,
i.e., fh ∈ B(g,u) ∩ I and thus g ∈ clI, which completes the proof. �

The next example shows that the converse of the above proposition is not true
in general.

Example 2.7. Consider the zero-dimensional space X = Q×Q, p = (0, 0) ∈ X,
and put Op = {f ∈ C(X) : p ∈ intXZ(f)} (note, Cc(X) = C(X) because X is
countable). Recall that Op is a zc-ideal. We now claim that Op is not a closed

ideal in C(X). To see this, consider f(x,y) =
|x|+|y|

1+|x|+|y| ∈ C(X) and let u be a
fixed positive unit of C(X). Define a function g by

g(x,y) =

{
0 where f(x,y) ≤ u(x,y)

2
,

f(x,y) − u(x,y)
2

where f(x,y) ≥ u(x,y)
2

.

Obviously, g ∈ C(X). Let G = {(x,y) ∈ X : f(x,y) < u(x,y)
2
}. Then

p ∈ G ⊆ Z(g) and therefore g ∈ Op, in fact, g ∈ B(f,u) ∩ Op. It follows
that f ∈ clC(X)Op. On the other hand, the set Z(f) = {p} is not open in X.
Hence, f ∈ clC(X)Op \Op. i.e., Op is not a closed ideal in C(X).

A Banach algebra B is an algebra that is a Banach space with a norm that
satisfies ‖xy‖≤ ‖x‖‖y‖ for all x,y ∈ B, and there exists a unit element e ∈ B
such that ex = xe = x, ‖e‖ = 1.

In [7, Definition 2.2], a topological space X is called a countably uniform
closed-space, briefly, a CUC-space, if whenever {fn}n∈N is a sequence of func-
tions of Cc(X) and fn → f uniformly, then f belongs to Cc(X).

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 83



A. Veisi

Theorem 2.8. If X is a CUC-space, then C∗c (X) with the supremum-norm
topology is a Banach algebra.

Proof. Let {fn}n∈N be a Cauchy sequence of functions in C∗c (X). Given ε > 0,
we can find a natural number N such that ‖fn −fm‖≤ ε for every m,n > N.
Thus, |fn(x) −fm(x)| ≤ ε for all x ∈ X and all m,n > N. Let x ∈ X be fixed
and ax be the limit of the numerical sequence {fn(x)}n∈N in R (note, R is a
Banach space). Now, define f : X → R by f(x) = ax. Let n be fixed, then
|fn(x)−limm→∞fm(x)| ≤ ε for each x ∈ X and each m > N. So ‖fn−f‖≤ ε.
Since n is arbitrary, we get fn → f in the norm, uniformly. Consequently,
f ∈ C(X). Furthermore, our assumption implies that f ∈ Cc(X). Moreover,
‖f‖ ≤ ‖f −fn‖ + ‖fn‖ gives f is bounded. Hence, C∗c (X) is a Banach space.
The proof is completed by the fact that ‖fg‖≤‖f‖‖g‖ for all f,g ∈ C∗c (X). �

3. Closed ideals in Cc(X) and C
∗
c (X) (with the mc-topology)

We need the next statement which is the counterpart of [5, 1D(1)] for Cc(X).

Proposition 3.1. If f,g ∈ Cc(X) and Z(f) is a neighborhood of Z(g), then
f = gh for some h ∈ Cc(X).

Proposition 3.2. Let X be a zero-dimensional space, f ∈ Cc(β0X) and let f0
be the restriction of f on X. Then intβ0XZ(f) ⊆ clβ0XZ(f0) ⊆ Z(f).

Proof. Let p ∈ intβ0XZ(f) and V be an open set in β0X containing p. Since
X is dense in β0X, we have ∅ 6= V ∩ intβ0XZ(f) ∩ X ⊆ V ∩ Z(f0). So
p ∈ clβ0XZ(f0). For the second inclusion, since Z(f0) ⊆ Z(f), we have that
clβ0XZ(f0) ⊆ clβ0XZ(f) = Z(f). �

Corollary 3.3. Let X be zero-dimensional and p ∈ β0X. Then
(i)
⋂
f∈Mpc clβ0XZ(f) = {p}.

(ii) If p ∈ X, then
⋂
f∈Mcp Z(f) = {p}, i.e., Mcp is fixed.

Proof. (i). Recall that f ∈ Mpc if and only if p ∈ clβ0XZ(f) (see [1, Theorem
4.2]). Therefore, p ∈

⋂
f∈Mpc clβ0XZ(f). Now, we claim that the latter intersec-

tion is the singleton set {p}. On the contrary, suppose that this set contains an
element q ∈ β0X distinct from p. Since β0X is zero-dimensional, by [3, Propo-
sition 4.4], there exists g ∈ Cc(β0X) such that p ∈ intβ0XZ(g) and g(q) = 1.
Let g0 be the restriction of g on X. Then by Proposition 3.2, clβ0XZ(g0) con-
tains p but not q. This means that g0 ∈ Mpc \Mqc which is a contradiction, so
(i) holds. (ii). Clearly,

⋂
f∈Mcp Z(f) =

⋂
f∈Mcp clβ0XZ(f) ∩X = {p}. �

In a similar way to Proposition 3.2 and Corollary 3.3, we get:

Proposition 3.4. For a Tychonoff space X and f ∈ C∗(X), we have that
intβXZ(f

β) ⊆ clβXZ(f) ⊆ Z(fβ), where fβ is the extension of f to βX.
Moreover, if p ∈ βX, then

⋂
f∈Mp clβXZ(f) = {p}. In particular, if p ∈ X,

then
⋂
f∈Mp Z(f) = {p}, i.e., Mp is fixed.

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 84



Closed ideals in the functionally countable subalgebra of C(X)

Proposition 3.5. Let X be zero-dimensional, p ∈ β0X and πp be its cor-
responding point of βX in characterizing of maximal ideals in Cc(X). Then
Mpc ∩C∗c (X) ⊆ M∗πp ∩C∗c (X). Particularly, if X is strongly zero-dimensional,
then Mpc ∩C∗c (X) ⊆ M∗p ∩C∗c (X).

Proof. In view of [1, Theorems 4.2, 4.8], we have

Mpc = {f ∈ Cc(X) : p ∈ clβ0XZ(f)} = {f ∈ Cc(X) : πp ∈ clβXZ(f)}.
Let f ∈ Mpc ∩C∗c (X). Then πp ∈ clβXZ(f) and hence fβ(πp) = 0, by Propo-
sition 3.4. Therefore, f ∈ M∗πp ∩ C∗c (X). The second part follows from the
assumption, i.e., β0X = βX and so πp = p. �

Remark 3.6. Replacing T with β0X in [1, Proposition 3.2] implies that for any
two zero-sets Z1 and Z2 in Zc(X), we get clβ0X(Z1 ∩Z2) = clβ0XZ1 ∩clβ0XZ2.

Remark 3.7. ([1, Remark 4.12]) If X is zero-dimensional and f,g ∈ Cc(X), then
clβ0XZ(f) is a neighborhood of clβ0XZ(g) if and only if there exists h ∈ Cc(X)
such that Z(g) ⊆ coz(h) ⊆ Z(f).

Proposition 3.8. Let X be zero-dimensional and I a proper ideal in Cc(X)
and let Vc(I) = {p ∈ β0X : Mpc ⊇ I}. Then:

(i) Vc(I) =
⋂
g∈I clβ0XZ(g).

(ii) If f ∈ Cc(X) and clβ0XZ(f) is a neighborhood of Vc(I), then f ∈ I.

Proof. (i). This is easily obtained from the fact that g ∈ Mpc if and only if
p ∈ clβ0XZ(g). (ii). Suppose that

Vc(I) =
⋂
g∈I

clβ0XZ(g) ⊆ intβ0Xclβ0XZ(f).

Then we have
⋃
g∈I
(
β0X \ clβ0XZ(g)

)
⊇ β0X \ intβ0Xclβ0XZ(f). Hence, the

collection

C = {intβ0Xclβ0XZ(f), β0X \ clβ0XZ(g) : g ∈ I}

is an open cover for the compact set β0X. Therefore, there is a finite number
of elements of I; g1,g2, . . . ,gn say, such that

β0X = intβ0Xclβ0XZ(f) ∪
(
β0X \ intβ0Xclβ0XZ(f)

)
= intβ0Xclβ0XZ(f) ∪

( n⋃
i=1

(β0X \ clβ0XZ(gi))
)
.

Now, we have that(⋂n
i=1 clβ0XZ(gi)

)
∩
(
β0X \ intβ0Xclβ0XZ(f)

)
= ∅.

Thus,
⋂n
i=1 clβ0XZ(gi) ⊆ intβ0Xclβ0XZ(f). Since I is a proper ideal, the ele-

ment g =
∑n
i=1 g

2
i of I is not a unit of Cc(X) and hence Z(g) =

⋂n
i=1 Z(gi) 6= ∅.

From Remark 3.6 we conclude that

clβ0XZ(g) = clβ0X
(⋂n

i=1 Z(gi)
)

=
⋂n
i=1 clβ0XZ(gi) ⊆ intβ0Xclβ0XZ(f).

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 85



A. Veisi

This leads us clβ0XZ(f) is a neighborhood of clβ0XZ(g). In view of Remark
3.7, there exists h ∈ Cc(X) such that Z(g) ⊆ coz(h) ⊆ Z(f). So Z(f) is a
neighborhood of Z(g). By Proposition 3.1, we get f ∈ I. �

Lemma 3.9. Let X be zero-dimensional and g ∈ Cc(X). Then for any neigh-
borhood B(g,u) of g in the mc-topology, there exists some fu ∈ B(g,u) such
that clβ0XZ(fu) is a neighborhood of clβ0XZ(g).

Proof. If clβ0XZ(g) is an open set in β0X, then we set fu = g. In general, we
define a function fu : X → R by

fu(x) =




g(x) − u(x)
2

where g(x) ≥ u(x)
2
,

0 where |g(x)| ≤ u(x)
2
,

g(x) +
u(x)

2
where g(x) ≤−u(x)

2
.

It is clear that fu ∈ C(X) and further since the range of g and u is countable,
we get fu ∈ Cc(X). Moreover, fu ∈ B(g,u). To establish the conclusion,
consider the function h below

h(x) =

{ (
g(x) +

u(x)
2

)(
g(x) − u(x)

2

)
where |g(x)| ≤ u(x)

2
,

0 where |g(x)| ≥ u(x)
2
.

We observe that h ∈ Cc(X). Furthermore, Z(g) ⊆ coz (h) ⊆ Z(fu). Now,
Remark 3.7 implies that clβ0XZ(fu) is a neighborhood of clβ0XZ(g), and we
are through. �

Theorem 3.10. Let X be zero-dimensional and I a proper ideal in Cc(X) and
let Vc(I) be the same as the set in Proposition 3.8

(
Vc(I) =

⋂
g∈I clβ0XZ(g)

)
.

Let

J = {f ∈ Cc(X) : clβ0XZ(f) ⊇ Vc(I)}, and Ī = ∩{M
p
c : M

p
c ⊇ I}.

Then:

(i) Ī is a closed ideal in Cc(X) containing I.
(ii) J = Ī, in other words, J is the kernel of the hull of I in the structure

space of Cc(X).
(iii) Vc(I) = Vc(Ī).
(iv) cl I = Ī.

Proof. (i). It follows from Corollary 2.2. (ii). Let f ∈ J and Mpc (p ∈ β0X) be
a maximal ideal in Cc(X) containing I. Then

(3.1) Vc(I) ⊇ Vc(Mpc ) and so clβ0XZ(f) ⊇ Vc(I) ⊇ Vc(M
p
c ) = {p}

(note, the last equality follows from Corollary 3.3). Therefore, f ∈ Mpc and
thus f ∈ Ī, i.e., J ⊆ Ī. For the reverse inclusion, we show that if f /∈ J, then
f /∈ Ī. Since f /∈ J, there exists q ∈ β0X such that q ∈ Vc(I) \ clβ0XZ(f).
Therefore, g ∈ Mqc for every g ∈ I and hence I ⊆ Mqc . But f /∈ Mqc . Thus,
Mqc is a maximal ideal containing I but not f. This yields that f /∈ Ī. (iii).
Using (ii) and the definition of J, we have Vc(Ī) = Vc(J) ⊇ Vc(I). On the
other hand, the inclusion I ⊆ Ī implies that Vc(Ī) ⊆ Vc(I). So (iii) holds.

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 86



Closed ideals in the functionally countable subalgebra of C(X)

(iv). By (i), clI ⊆ Ī. Now, suppose that g ∈ Ī and u is a positive unit
of Cc(X). We claim that B(g,u) ∩ I 6= ∅. According to Lemma 3.9, there
exists fu ∈ Cc(X) such that fu ∈ B(g,u), and clβ0XZ(fu) is a neighborhood
of clβ0XZ(g). Now, it remains to show that fu ∈ I. From (iii), we infer that
Vc(I) = Vc(Ī) ⊆ clβ0XZ(g) ⊆ intβ0Xclβ0XZ(fu). Proposition 3.8(ii) now yields
that fu ∈ I. Therefore, fu ∈ B(g,u) ∩ I and so g ∈ clI, i.e., Ī ⊆ clI. �

It is known that a proper ideal in C(X) with the m-topology is closed if and
only if it is an intersection of maximal ideals in C(X) (see [5, 7Q(2)]). The
next theorem involves the countable analogue characterization of closed ideals
in Cc(X). Using Theorem 3.10(iv) and Corollary 2.2, we obtain:

Theorem 3.11. Let X be zero-dimensional and the topology on Cc(X) be the
mc-topology. Then a proper ideal in Cc(X) is closed if and only if it is an
intersection of maximal ideals of Cc(X).

Theorem 3.12. Let X be zero-dimensional and the topology on Cc(X) (resp.
C(X)) be the mc-topology (resp. the m-topology). Then the following state-
ments are equivalent.

(i) Every ideal in C(X) is closed.
(ii) X is a P -space.
(iii) Every ideal in Cc(X) is closed.
(iv) Every prime ideal in Cc(X) is closed.

Proof. (i) ⇔ (ii). It follows from [5, 4J(9), 7Q(2)].
(ii) ⇒ (iii). By [3, Proposition 5.3], X is a CP-space. Now, the result is

obtained by [3, Theorem 5.8(7)] and Corollary 2.2.
(iii) ⇒ (iv). It is evident.
(iv) ⇒ (ii). According to [3, Corollary 5.7], it is enough to show that X is a

CP-space. Let P be a prime ideal in Cc(X), then by [1, Lemma 4.11(4)], P is
contained in a unique maximal ideal Mpc of Cc(X), where p ∈ β0X. Now, by
the assumption and Theorem 3.11, we get P = Mpc , i.e., X is a CP-space. �

Theorem 3.13. Let X be strongly zero-dimensional and the topology on C∗c (X)
(resp. C∗(X)) be the mc-topology (resp. the m-topology). Then the following
statements are equivalent.

(i) Every properly closed ideal in C∗c (X) is an intersection of maximal
ideals of C∗c (X).

(ii) X is pseudocompact.
(iii) Every properly closed ideal in C∗(X) is an intersection of maximal

ideals of C∗(X).

Proof. A maximal ideal in C∗c (X) is of the form M
∗p
c = {f ∈ C∗c (X) : fβ(p) = 0},

where p ∈ βX. Also, M∗pc = M∗p ∩C∗c (X), see [9, Corollaries 2.10, 2.11].
(i) ⇒ (ii). Suppose that X is not pseudocompact, so C∗c (X) $ Cc(X), by

[9, Theorem 6.3]. Hence, Cc(X) contains an unbounded element, f say. So for
some p ∈ βX and the maximal ideal Mpc of Cc(X), we have |Mpc (f)| is infinitely

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 87



A. Veisi

large ([9, Proposition 2.4]). In other words, Mpc is hyper-real, i.e., R $
Cc(X)
M

p
c

.

Hence, by [9, Corollary 2.13], Mpc ∩ C∗c (X) is not a maximal ideal in C∗c (X).
Using Proposition 3.5, we infer that

(3.2) Mpc ∩C
∗
c (X) $ M

∗p ∩C∗c (X).

Furthermore, since the maximal ideal Mpc is closed in Cc(X) (Corollary 2.2),
the ideal Mpc ∩C∗c (X) is also closed in C∗c (X). We now claim that the latter
closed ideal cannot be an intersection of maximal ideals of C∗c (X). Otherwise,

(3.3) Mpc ∩C
∗
c (X) =

⋂
q∈A⊆βX

(
M∗q ∩C∗c (X)

)
,

for a subset A of βX. Notice that by (3.2), A 6= ∅ since p ∈ A. Now, we
claim that A = {p}. On the contrary, suppose that A contains an element
q distinct from p. We can take f ∈ Cc(βX) such that Z(f) is a neighbor-
hood of p and f(q) = 1 (note, by the assumption, βX is zero-dimensional).
Let f0 be the restriction of f on X. Then the compactness of βX gives f
and hence f0 are bounded, i.e., f0 ∈ C∗c (X). By density of X in βX, we get
f = f

β
0 , where f

β
0 is the extension of f0 to βX. Due to Proposition 3.2, we

infer that p ∈ clβXZ(f0), since p ∈ intβXZ(f). Hence, f0 ∈ Mpc ∩ C∗c (X).
On the other hand, since q /∈ Z(f), we have that f0 /∈ M∗q. Therefore,
f0 ∈ Mpc ∩C∗c (X) \ (M∗q ∩C∗c (X)), which contradicts the equation in (3.3).
So A = {p} and hence Mpc ∩C∗c (X) = M∗p ∩C∗c (X). But this also contradicts
(3.2). Thus, if X is not pseudocompact, then there exists a closed ideal in
C∗c (X) which is not an intersection of maximal ideals of C

∗
c (X), and we are

done.
(ii) ⇒ (i). Since X is pseudocompact, C(X) = C∗(X) gives Cc(X) = C∗c (X).

Now, it follows from Theorem 3.11.
(ii) ⇔ (iii). It follows from [5, 7Q(3)]. �

We end the article with some results on ec-filters on X and ec-ideals in
C∗c (X), for more details, see [14, Section 2]. Let p ∈ βX and fβ be the
extension of f ∈ C∗(X) to βX. Let us recall that

M∗pc = {f ∈ C
∗
c (X) : f

β(p) = 0} = M∗p ∩C∗c (X), and O
∗p
c = O

p
c ∩C

∗
c (X),

where

M∗p = {f ∈ C∗(X) : fβ(p) = 0}, and Opc = {f ∈ Cc(X) : p ∈ intβXclβXZ(f)}.

Lemma 3.14. Let X be strongly zero-dimensional and p ∈ βX. Then
Ec(M

∗p
c ) = Zc[O

p
c ] = Zc[O

∗p
c ] = Ec(O

∗p
c ).

Proof. By the hypothesis, βX = β0X. To get the result, we show the following
chain of inclusions holds.

(3.4) Ec(M
∗p
c ) ⊆ Zc[O

p
c ] ⊆ Zc[O

∗p
c ] ⊆ Ec(O

∗p
c ) ⊆ Ec(M

∗p
c ).

To establish the first inclusion, let Ecε(f) := {x ∈ X : |f(x)| ≤ ε} ∈ Ec(M∗pc ),
where f ∈ M∗pc and ε > 0. Then fβ(p) = 0. Notice that Ecε(f) = Z((|f|−ε)∨0)

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 88



Closed ideals in the functionally countable subalgebra of C(X)

and

(3.5) clβXZ((|f|−ε) ∨ 0) = clβXEcε(f) = {q ∈ βX : |f
β(q)| ≤ ε}.

Hence, p ∈ intβXclβXZ((|f|−ε) ∨ 0), in other words, (|f|−ε) ∨ 0 ∈ Opc .
Here, we are going to show the last equality in (3.5). Let q ∈ βX such that
|fβ(q)| ≤ ε. Since X is dense in βX, there exists a net (xλ)λ∈Λ ⊆ X converging
to q and so f(xλ) = f

β(xλ) → fβ(q). Moreover, |f(xλ)| → |fβ(q)|. Now, let
V be an open set in βX containing q. Then for some λ0 ∈ Λ and each λ ≥ λ0,
we have xλ ∈ V . Furthermore, |fβ(q)| ≤ ε yields that |f(xλ)| ≤ ε. Hence,
V ∩Ecε(f) 6= ∅, i.e., q ∈ clβXEc� (f).

The second inclusion in (3.4) follows from the fact that Z(f) = Z( f
1+|f|),

where f ∈ Opc (and thus
f

1+|f| ∈ O
∗p
c ). To verify the third inclusion, we

let f ∈ O∗pc and show that Z(f) ∈ Ec(O∗pc ). Since p does not belong to
the closed set F := βX \ intβXclβXZ(f) and βX is zero-dimensional, by [3,
Proposition 4.4], there is some g ∈ Cc(βX) = C∗c (βX) such that p ∈ intβXZ(g)
and g(F) = {1}. Let g0 be the restriction of g on X. Then by Proposition
3.2, p ∈ intβXclβXZ(g0). So g0 ∈ O∗pc and hence Ecε(g0) ∈ Ec(O∗pc ) for all
ε > 0. Let 0 < ε < 1 be fixed. Since X is dense in βX, the open set
{q ∈ βX : |g(q)| < ε} intersects X nontrivially (since it contains p). Therefore,

∅ 6= {q ∈ βX : |g(q)| ≤ ε}∩X = {x ∈ X : |g0(x)| ≤ ε}
= Ecε(g0) ⊆ (βX \F) ∩X ⊆ Z(f).

Now, since the zc-filter (in fact, the ec-filter) Ec(O
∗p
c ) contains E

c
ε(g0) and

Ecε(g0) ⊆ Z(f), we infer that Z(f) ∈ Ec(O∗pc ), and we are done.
Finally, the last inclusion in (3.4) follows from the inclusion O∗pc ⊆ M∗pc and
the fact that Ec preserves the order, see [14, Corollary 2.1]. �

Theorem 3.15. Let X be a P -space and F, an ec-filter on X. Then F is an
ec-ultrafilter if and only if it is a zc-ultrafilter.

Proof. (⇒) : By [5, 4K(7), 6M(1), 16O], every P-space is strongly zero- dimen-
sional (see also [15, Proposition 2.12]). By [5, 7L], we have Op = Mp for every
p ∈ βX. Therefore, Opc = Op∩Cc(X) = Mp∩Cc(X) = Mpc (note, βX = β0X).
Let F be an ec-ultrafilter on X. Then E−1c (F) is a maximal ideal in C∗c (X),
see [14, Proposition 2.14]. Therefore, E−1c (F) = M∗pc for some p ∈ βX. By
Lemma 3.14, we have

F = Ec(E−1c (F)) = Ec(M
∗p
c ) = Zc[O

p
c ] = Zc[M

p
c ].

Since Mpc is a maximal ideal in Cc(X), F is a zc-ultrafilter.
(⇐) : Suppose that F is a zc-ultrafilter. Then Z−1c [F] is a maximal ideal in
Cc(X). So Z

−1
c [F] = Mpc for some p ∈ βX. Therefore,

F = Zc[Z−1c [F]] = Zc[M
p
c ] = Ec(M

∗p
c ).

Since M∗pc is a maximal ideal in C
∗
c (X), F is an ec-ultrafilter. �

Corollary 3.16. For a strongly zero-dimensional space X and p ∈ βX, M∗pc
is the only ec-ideal in C

∗
c (X) containing O

∗p
c .

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 89



A. Veisi

Proof. Let J be an ec-ideal in C
∗
c (X) which contains O

∗p
c . Then

E−1c (Ec(O
∗p
c )) ⊆ E−1c (Ec(J)) = J. By Lemma 3.14, Ec(M∗pc ) = Ec(O∗pc ) and

therefore
M∗pc = E

−1
c (Ec(M

∗p
c )) = E

−1
c (Ec(O

∗p
c )) ⊆ J.

So M∗pc = J, and we are through. �

Acknowledgements. The author is grateful to the referee for useful com-
ments and recommendations towards the improvement of the paper.

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