@ Appl. Gen. Topol. 22, no. 2 (2021), 295-302doi:10.4995/agt.2021.13696
© AGT, UPV, 2021

Intrinsic characterizations of c-realcompact

spaces

Sudip Kumar Acharyya, Rakesh Bharati and A. Deb Ray

Department of Pure Mathematics, University of Calcutta, 35, Ballygunge Circular Road, Kolkata -

700019, INDIA (sdpacharyya@gmail.com, bharti.rakesh292@gmail.com, debrayatasi@gmail.com)

Communicated by O. Valero

Abstract

c-realcompact spaces are introduced by Karamzadeh and Keshtkar in
Quaest. Math. 41, no. 8 (2018), 1135–1167. We offer a characteriza-
tion of these spaces X via c-stable family of closed sets in X by showing
that X is c-realcompact if and only if each c-stable family of closed sets
in X with finite intersection property has nonempty intersection. This
last condition which makes sense for an arbitrary topological space can
be taken as an alternative definition of a c-realcompact space. We show
that each topological space can be extended as a dense subspace to a
c-realcompact space with some desired extension properties. An allied
class of spaces viz CP-compact spaces akin to that of c-realcompact
spaces are introduced. The paper ends after examining how far a known
class of c-realcompact spaces could be realized as CP-compact for ap-
propriately chosen ideal P of closed sets in X.

2010 MSC: 54C40.

Keywords: c-realcompact spaces; Banaschewski compactification; c-stable
family of closed sets; ideals of closed sets; initially θ-compact
spaces.

1. Introduction

In what follows X stands for a completely regular Hausdorff topological space.
As usual C(X) and C∗(X) denote respectively the ring of all real valued contin-
uous functions on X and that of all bounded real valued continuous functions

Received 16 May 2020 – Accepted 10 May 2021

http://dx.doi.org/10.4995/agt.2021.13696


S. K. Acharyya, R. Bharati and A. Deb Ray

on X. Suppose Cc(X) is the subring of C(X) containing those functions f for
which f(X) is a countable set and C∗c (X) = Cc(X) ∩ C

∗(X). Formal inves-
tigations of these two rings vis-a-vis the topological structure of X are being
carried on only in the recent times. It turns out that there is an interplay
between the topological structure of X and the ring and lattice structure of
Cc(X) and C

∗
c (X), which incidentally sheds much light on the topology of X.

The articles [3], [4], [7], [8], [11] may be referred in this context. The notion
of c-realcompact spaces is the fruit of one such endeavours in the study of X
versus Cc(X) or C

∗
c (X). A space X is declared c-realcompact in [8] if each

real maximal ideal M in Cc(X) is fixed in the sense that there exits a point
x ∈ X such that for each f ∈ M, f(x) = 0. M is called real when the residue
class field Cc(X)/M is isomorphic to the field R. A number of interesting facts
concerning these spaces is discovered in [8]. These may be called countable
analogues of the corresponding properties of real compact spaces as developed
in [6], chapter 8. In the present article we offer a new characterization of c-
realcompact spaces on using the notion, c-stable family of closed sets in X. A
family F of subsets of X is called c-stable if given f ∈ C(X,Z), there exists
F ∈ F such that f is bounded on F .

We define a topological space X (not necessarily completely regular ) to be
cc-realcompact if each c-stable family of closed sets in X with finite intersec-
tion property has nonempty intersection. We check that this new notion of cc-
realcompactness agrees with the already introduced notion of c-realcompactness
in [8], within the class of zero-dimensional Hausdorff spaces (Theorem 2.3).
We re-establish a modified version of a few known properties of c-realcompact
spaces using our new definition of cc-realcompactness (Theorem 2.4). Fur-
thermore we realize that any topological space X can be extended as a dense
subspace to a cc-realcompact space υ0X enjoying some desired extension prop-
erties (Theorem 2.5). While constructing this extension of X, we follow closely
the technique adopted in [9]. The results mentioned above constitute the first
technical section viz §2 of this article.

A family P of closed sets in X is called an ideal of closed sets if A ∈ P, B ∈ P
and C is a closed subset of A imply that A ∪ B ∈ P and C ∈ P. Let Ω(X)
stand for the aggregate of all ideals of closed sets in X. For any P ∈ Ω(X) let
CP(X) = {f ∈ C(X) : clX(X \ Z(f)) ∈ P}, here Z(f) = {x ∈ X : f(x) = 0}
is the zero set of f in X. It is well known that CP(X) is an ideal in the ring
C(X), see [1] and [2] for more information on these ideals. With reference
to any such P ∈ Ω(X), we call a family F of subsets of X cP-stable if given
f ∈ C(X,Z) ∩ CP(X) there exists F ∈ F such that f is bounded on F . We
define a space X to be cP-compact if any cP-stable family of closed sets in X
with finite intersection property has non-empty intersection. It is clear that a
zero-dimensional space X is cc-realcompact if it is already cP-compact.

We have shown that if X is a noncompact zero-dimensional space and P ∈
Ω(X) such that X is cP-compact, then there exists an R ∈ Ω(X) such that R &
P and X is cR-compact. Thus within the class of zero-dimensional noncompact
spaces X, there is no minimal member P ∈ Ω(X) in the set inclusion sense of

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Intrinsic characterizations of c-realcompact spaces

the term for which X becomes cP-compact (Theorem 3.2). In the concluding
portion of §3 of this article we have examined, how far the known classes of
c-realcompact spaces could be achieved as cP-compact spaces for appropriately
chosen P ∈ Ω(X). For any infinite cardinal number θ, X is called finally
θ-compact if each open cover of X has a subcover with cardinality < θ (see
[10]). In this terminology finally ω1-compact spaces are Lindelöf and finally
ω0-compact spaces are compact. It is realized that a c-realcompact space X
is finally θ-compact if and only if it is cQ-compact, where Q is the ideal of all
closed finally θ-compact subsets of X (Theorem 3.4). A special case of this
result reads: X is Lindelöf when and only when X is cα-compact where α is
the ideal of all closed Lindelöf subsets of X.

2. Properties of cc-realcompact spaces and
cc-realcompactifications

Before stating the first technical result of this section, we need to recall a
few terminologies and results from [4] and [8]. Our intention is to make the
present article self contained as far as possible. An element α on a totally
ordered field F is called infinitely large if α > n for each n ∈ N. It is clear that
F is archimedean if and only if it does not contain any infinitely large element.
If M is maximal ideal in Cc(X) then the residue class field Cc(X)/M is totally
ordered according to the following definition: for f ∈ C(X), M(f) ≧ 0 if and
only if there exists g ∈ M such that f ≧ 0 on Z(g). Here M(f) stands for the
residue class in C(X)/M, which contains the function f.

Theorem 2.1 (Proposition 2.3 in [8]). For a maximal ideal M in Cc(X) and
for f ∈ Cc(X), |M(f)| is infinitely large in Cc(X)/M if and only if f is
unbounded on every zero set of Zc(M) = {Z(g) : g ∈ M}.

It is proved in [4], Remark 3.6 that if X is a zero-dimensional space, then
the set of all maximal ideals of Cc(X) equipped with hull-kernel topology,
also called the structure space of Cc(X) is homeomorphic to the Banaschewski
compactification β0X of X. Thus the maximal ideals of Cc(X) can be indexed
by virtue of the points of β0X. Indeed a complete description of all these
maximal ideals is given by the list {Mpc : p ∈ β0X}, where M

p
c = {f ∈ Cc(X) :

p ∈ clβ0XZ(f)} with M
p
c is a fixed maximal ideal if and only if p ∈ X (see

Theorem 4.2 in [4]). It is well known that any continuous map f : X → Y ,
where X and Y are both zero-dimensional spaces with Y compact also, has
an extension to a continuous map f̄ : β0X → Y (we call this property, the
C-extension property of β0X) (see Remark 3.6 in [4]). It follows that for a
zero-dimensional space X, any continuous map f : X → Z (also written as
f ∈ C(X,Z)), has an extension to a continuous map f∗ : β0X → Z

∗ = Z∪{ω},
the one point compactification of Z. We also write f∗ ∈ C(β0X,Z

∗). A slightly
variant form of the next result is proved in [8], Theorem 2.17 and Theorem 2.18.

Theorem 2.2. Let X be zero-dimensional and p ∈ β0X, then the maximal
ideal Mpc in Cc(X) is real if and only if for each f ∈ C(X,Z), f

∗(p) 6= ω if
and only if |MPc (f)| is not infinitely large in Cc(X)/M

p
c .

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S. K. Acharyya, R. Bharati and A. Deb Ray

Theorem 2.3. A zero-dimensional space X is cc-realcompact if and only if it
is c-realcompact.

Proof. Let X be a c-realcompact space and F be a family of closed subsets
of X with finite intersection property but with

⋂
F=∅. To show that X is

cc-realcompact we shall prove that F is not a c-stable family. Indeed {clβ0XF :
F ∈ F} is a family of closed subsets of β0X with finite intersection property.
Since β0X is compact, there exists a point p ∈

⋂

F ∈F

clβ0XF and of course

p ∈ β0X \ X. Here M
p
c is a free maximal ideal in Cc(X). Since X is c-

realcompact this implies that Mpc is a hyperreal maximal ideal (meaning that
it is not a real maximal ideal of Cc(X)). It follows from Theorem 2.2 that there
exists f ∈ C(X,Z) with f∗(p) = ω. Since p ∈ clβ0XF for each F ∈ F, it is
therefore clear that ‘f’ is unbounded on each set in the family F. Therefore F
is not a c-stable family.

Conversely let X be not c-realcompact. Then there exists a real maximal
ideal M in Cc(X), which is not fixed. This means that there is a point p ∈
β0X \ X for which M = M

p
c . Since p ∈ clβ0XZ(f) for each f ∈ M

p
c , it follows

that {Z(f) : f ∈ Mpc } is a family of closed sets in X with finite intersection
property but with empty intersection. To show that X is not cc-realcompact, it
suffices to show that {Z(f) : f ∈ Mpc } is a c-stable family. So let g ∈ C(X,Z).
Since Mpc is real, this implies in view of Theorem 2.2 that g

∗(p) 6= ω and hence
|Mpc (g)| is not infinitely large. It follows therefore from Theorem 2.1 that g is
bounded on some Z(f) for an f ∈ Mpc . This settles that {Z(f) : f ∈ M

p
c } is a

c-stable family. �

By adapting the arguments of Theorem 5.2, Theorem 5.3 and Theorem 5.4
in [9] appropriately, we can establish the following facts about cc-realcompact
spaces without difficulty:

Theorem 2.4.

(1) A compact space is cc-realcompact.
(2) A pseudocompact cc-realcompact space is compact.
(3) A closed subspace of a cc-realcompact space is cc-realcompact.
(4) The product of any set of cc-realcompact spaces is cc-realcompact.
(5) If a topological space X = E ∪ F where E is a compact subset of X

and F is a Z-embedded cc-realcompact subset of X, meaning that each
function in C(F,Z) can be extended to a function in C(X,Z), then X
is cc-realcompact.

(6) A Z-embedded cc-realcompact subset of a Hausdorff space X is a closed
subset of X.

We now show that any topological space X can be extended to a cc-realcompact
space containing the original space X as a C-embedded dense subspace and en-
joying a desirable extension property. The proof can be accomplished by closely
following the arguments adopted to prove Theorem 6.1 in [9]. Nevertheless we
give a brief outline of the main points of proof in our theorem.

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Intrinsic characterizations of c-realcompact spaces

Theorem 2.5. Every topological space X can be extended to a cc-realcompact
space υcX as a dense subspace with the following extension property: each
continuous map from X into a regular cc-realcompact space Y can be extended
to a continuous map from υcX into Y . X is cc-realcompact if and only if
X = υcX.

Proof. For each x ∈ X let Gx be the aggregate of all closed sets in X which
contain the point x. Then Gx is a c-stable family of closed sets in X with finite
intersection property and with the prime condition: A∪B ∈ Gx =⇒ A ∈ Gx or
B ∈ Gx, A,B ⊆ X. We extend the set X to a bigger set υcX, so that υcX \ X
becomes an index set for the collection of all maximal c-stable families of closed
subsets of X with finite intersection property but with empty intersection. For
each p ∈ υcX\X, let Gp designate the corresponding maximal c-stable family of
closed sets in X with finite intersection property and with empty intersection.
For each closed set F in X, we write F̄ = {p ∈ υcX : F ∈ Gp}. Then {F̄: F is
closed in X} forms a base for closed sets of some topology on υcX and in this
topology for any closed set F in X F̄ = clυcXF . Since X belongs to each G

p, it
is clear that X is dense in υcX. Let t : X → Y be a continuous map with Y , a
regular cc-realcompact space. Choose p ∈ υ

cX. Let Hp = {G ⊆ Y : G is closed
in Y and t−1(G) ∈ Gp}. Then Hp is a c-stable family of closed sets in Y with
finite intersection property. We select a point y ∈

⋂
Hp and we set t0(p) = y

with the aggrement that t0(p) = t(p) in case p ∈ X. Thus t0 : υcX → Y
is a well defined map which is further continuous. The remaining parts of the
theorem can be proved by making arguments closely as in the proof of Theorem
6.1 of [9]. �

3. cP-compact spaces

In this section all the topological spaces X that will appear will be assumed
to be zero-dimensional. We define for any P ∈ Ω(X), υP0 (X) = {p ∈ β0X :
f∗(p) 6= ω for each f ∈ CP(X) ∩ C(X,Z)}. It is clear that if P=E≡ the ideal
of all closed sets in X then υE0 (X) = υ0X ≡ {p ∈ β0X : f

∗(p) 6= ω for each
f ∈ C(X,Z)} the set defined in the begining of the proof of Theorem 3.8 in
[8]. The next theorem puts Theorem 2.3 in a more general setting.

Theorem 3.1. For a P ∈ Ω, X is cP-compact if and only if X = υ
P
0 (X).

We omit the proof of this theorem because it can be done by making some
appropriate modification in the arguments adopted in the proof of Theorem
2.3.
It is clear that if P, Q ∈ Ω(X) with P ⊂ Q, then any cQ-stable family of
closed sets in X is also cP-stable, consequently if X is cP-compact then X is
cQ-compact also. In particular every cP-compact space is cc-realcompact and
hence c-realcompact in view of Theorem 2.3. The following question therefore
seems to be natural.
If X is a zero-dimensional non-compat c-realcompact space, then does there
exist a minimal ideal P of closed sets in X (minimal in some sense of the
term) for which X becomes cP-compact?

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S. K. Acharyya, R. Bharati and A. Deb Ray

No possible answer to this question is known to us, however the following
proposition shows that the answer to this question is in the negative if the
phrase ‘minimal’ is interpreted in the set inclusion sense of the term.

Theorem 3.2. Let X be a non compact zero-dimensional space. Suppose P ∈
Ω(X) is such that X is cP-compact. Then there exists R ∈ Ω(X) such that
R & P and X is cR-compact.

Proof. We get from Theorem 3.1 that X = υP0 X. As X is non compact we
can choose a point p ∈ β0X \ X. Then p /∈ υ

P
0 X. Accordingly there exists

f ∈ CP(X) ∩ C(X,Z) such that f
∗(p) = ω. We select a point x ∈ X such

that f(x) 6= 0. Set R = {D ∈ P : x /∈ D}. It is easy to check that R is an
ideal of closed sets in X, i.e., R ∈ Ω(X). Furthermore, clX(X − Z(f)) is a
member of P containing the point x. This implies that clX(X − Z(f)) /∈ R.
Thus R  P. To show that X is cR-compact. We shall show that X = υ

R
0 X

(see Theorem 3.1). So choose a point q ∈ β0X \X then q /∈ υ
P
0 X, consequently

there exists g ∈ CP(X) ∩ C(X,Z) such that g
∗(q) = ω. For the distinct points

q,x in β0X there exist disjoint open sets U, V in this space such that x ∈ U,
q ∈ V . Since β0X is zero-dimensional there exists therefore a clopen set W in
β0X such that q ∈ W ⊂ V . The map h : β0X → {0,1} given by h(W) = {1}
and h(β0X \ W) = {0} is continuous. We note that h(U) = {0} and h(q) = 1.
Let ψ = h|X. Then ψ ∈ C(X,Z). Take l = g.ψ. Since g ∈ CP(X) and CP(X)
is an ideal of C(X), it follows that l ∈ CP(X). Furthermore the fact that g and
ψ are both functions in C(X,Z) implies that l ∈ C(X,Z). Also the function
h ∈ C(β0X,Z) is the unique continuous extension of ψ ∈ C(X,Z), hence we
can write h = ψ∗. This implies that l∗(q) = g∗(q)ψ∗(q) = g∗(q)h(q) = ω,
because g∗(q) = ω and h(q) 6= 0.
On the other hand if y ∈ U∩X then h(y) = 0 and hence l(y) = 0. Since U∩X is
an open neighbourhood of x in the space X, this implies that x /∈ clX(X\Z(l)).
Since clX(X \ Z(l)) ∈ P, already verified, it follows that clX(X \ Z(l)) ∈ R.
Thus l ∈ CR(X) ∩ C(X,Z). Since l

∗(q) = ω, this further implies that q /∈
υR0 (X). �

It is trivial that a (zero-dimensional) compact space is c-realcompact. It is
also observed that a Lindelöf space is c-realcompact (Corollary 3.6, [8]). But
for an infinite cardinal number θ, a finally θ-compact space may not be c-
realcompact. Indeed the space [0,ω1) of all countable ordinals is a celebrated
example of a zero-dimensional space which is not realcompact (see 8.1, [6]).
Since a zero-dimensional c-realcompact space is necessarily realcompact (vide
proposition 5.8, [8]) it follows therefore that [0,ω1) is not a c-realcompact space.
But it is easy to show that [0,ω1) is finally ω2-compact. For the same reason,
the Tychonoff plank T ≡[0,ω1)×[0,ω0)-{(ω1,ω0)} of 8.20 in [6], is finally ω2-
compact without being c-realcompact. It can be easily shown that a closed
subset of a finally θ-compact space is finally θ-compact.

Furthermore, the following characterization of finally θ-compactness of a
topological space can be established by routine arguments.

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 300



Intrinsic characterizations of c-realcompact spaces

Theorem 3.3. The following two statements are equivalent for an infinite
cardinal number θ.

(1) X is finally θ-compact.
(2) If B is a family of closed sets in X, such that for any subfamily B0 of

B with |B0| < θ,
⋂
B0 6= ∅, then

⋂
B 6= ∅.

Theorem 3.4. Let X be c-realcompact and Pθ the ideal of all closed finally
θ-compact subsets of X. Then X is finally θ-compact if and only if it is cPθ -
compact.

Proof. Let X be finally θ-compact and p ∈ β0X \ X. To show that X is cPθ -

compact, it suffices to show in view of Theorem 3.1 that p /∈ υPθ0 (X). Indeed
X is c-realcompact implies that the maximal ideal Mpc of Cc(X) is not real.
Consequently by Theorem 2.2, there exists f ∈ C(X,Z) such that f∗(p) = ω.
Now clX(X \ Z(f)), like any closed subsets of X is finally θ-compact. Thus

f ∈ CPθ (X) ∩ C(X,Z), hence p /∈ υ
Pθ
0 (X).

To prove the converse, let X be not finally θ-compact. It follows from Theorem
3.3 that there exists a family B = {Bα : α ∈ Λ} of closed sets in X with the
following properties: for any subfamily B1 of B with |B1| < θ,

⋂
B1 6= ∅ but⋂

B = ∅. Let D = {Dα : α ∈ Λ
∗} be the aggregate of all sets D′αs, which

are intersections of < θ many sets in the family B. Then B ⊆ D and hence
⋂
D = ∅. Also D has finite intersection property. We shall show that D is a

cPθ-stable family and hence X is not Pθ-compact. Towards such a proof choose
f ∈ CPθ (X) ∩ C(X,Z), then clX(X \ Z(f) is a finally θ-compact subset of X.
Since {X \ Bα : α ∈ Λ} is an open cover of X, there exists a subset Λ0 of
Λ with |Λ0| < θ such that clX(X \ Z(f)) ⊆

⋃

α∈Λ0

(X \ Bα). This implies that
⋂

α∈Λ0

Bα ⊆ Z(f) and we note that
⋂

α∈Λ0

Bα ∈ D. Thus f becomes bounded on

a set lying in the family D. Hence D becomes a cPθ-stable family. �

Acknowledgements. The second author acknowledges financial support from
University Grand Commission, New Delhi, for the award of research fellowship
(F. No. 16-9(June 2018)/2019 (NET/CSIR)).

References

[1] S. K. Acharyya and S. K. Ghosh, A note on functions in C(X) with support lying on
an ideal of closed subsets of X, Topology Proc. 40 (2012), 297–301.

[2] S. K. Acharyya and S. K. Ghosh, Functions in C(X) with support lying on a class of
subsets of X, Topology Proc. 35 (2010), 127–148.

[3] S. K. Acharyya, R. Bharati and A. Deb Ray, Rings and subrings of continuous functions
with countable range, Queast. Math., to appear.

[4] F. Azarpanah, O. A. S. Karamzadeh, Z. Keshtkar and A. R. Olfati, On maximal ideals
of Cc(X) and the uniformity of its localizations, Rocky Mountain J. Math. 48, no. 2
(2018), 345–384.

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 301



S. K. Acharyya, R. Bharati and A. Deb Ray

[5] P. Bhattacherjee, M. L. Knox and W. W. Mcgovern, The classical ring of quotients of
Cc(X), Appl. Gen. Topol. 15, no. 2 (2014), 147–154.

[6] L. Gillman and M. Jerison, Rings of Continuous Functions, Van Nostrand Reinhold co.,
New York, 1960.

[7] M. Ghadermazi, O. A. S. Karamzadeh and M. Namdari, On the functionally countable
subalgebras of C(X), Rend. Sem. Mat. Univ. Padova. 129 (2013), 47–69.

[8] O. A. S. Karamzadeh and Z. Keshtkar, On c-realcompact spaces, Queast. Math. 41, no.
8 (2018), 1135–1167.

[9] M. Mandelkar, Supports of continuous functions, Trans. Amer. Math. Soc. 156 (1971),
73–83.

[10] R. M. Stephenson Jr, Initially k-compact and related spaces, in: Handbook of Set-
Theoretic Topology, ed. Kenneth Kunen and Jerry E. Vaughan. Amsterdam, North-
Holland, (1984) 603–632.

[11] A. Veisi, ec-filters and ec-ideals in the functionally countable subalgebra of C
∗(X), Appl.

Gen. Topol. 20, no. 2 (2019), 395–405.

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 302