@
Appl. Gen. Topol. 23, no. 1 (2022), 1-16

doi:10.4995/agt.2022.15668

© AGT, UPV, 2022

Some classes of topological spaces related
to zero-sets

F. Golrizkhatami and A. Taherifar

Department of Mathematics, Yasouj University, Yasouj, Iran (F.golrizkhatami@stu.yu.ac.ir,

ataherifar@yu.ac.ir)

Communicated by D. Georgiou

Abstract

An almost P -space is a topological space in which every zero-set is
regular-closed. We introduce a large class of spaces, C-almost P -space
(briefly CAP -space), consisting of those spaces in which the closure
of the interior of every zero-set is a zero-set. In this paper we study
CAP -spaces. It is proved that if X is a dense and Z#-embedded sub-
space of a space T , then T is CAP if and only if X is a CAP and
CRZ-extended in T (i.e, for each regular-closed zero-set Z in X, clT Z
is a zero-set in T ). In 6P.5 of [8] it was shown that a closed count-
able union of zero-sets need not be a zero-set. We call X a CZ-space
whenever the closure of any countable union of zero-sets is a zero-set.
This class of spaces contains the class of P -spaces, perfectly normal
spaces, and is contained in the cozero complemented spaces and CAP -
spaces. In this paper we study topological properties of CZ (resp. coz-
ero complemented)-space and other classes of topological spaces near
to them. Some algebraic and topological equivalent conditions of CZ
(resp. cozero complemented)-space are characterized. Examples are
provided to illustrate and delimit our results.

2020 MSC: 54C40.

Keywords: zero-set; almost P-space; compact space; z-embedded subset.

1. Introduction

The set of zero-sets in a topological space X, Z[X], need not be closed under
infinite union. Even a countable union of zero-sets need not be a zero-set. For

Received 21 May 2021 – Accepted 20 September 2021

http://dx.doi.org/10.4995/agt.2022.15668


F. Golrizkhatami and A. Taherifar

example, every one-element set in R is a zero-set in R, but Q =
⋃
r∈Q{r} is

not a zero-set. First, we call a countable subfamily of Z[X] a CZ-family if the
union of its elements is a zero-set (cf. Definition 3.1). A question for us was:

When is any countable subfamily of Z[X] a CZ-family?

We observe that in a space X every countable subfamily is a CZ-family if and
only if X is a P-space (cf. Proposition 3.2). In 5.15 of [8], it is shown that if a
countable union of zero-sets belongs to a real z-ultrafilter A, then at least one
of them belongs to A. But we need the converse of this fact for our aims. It is
shown that for an ideal I of C(X),

⋃∞
i=1 Z(fi) ∈ Z[I] implies fi ∈ I for some

i ∈ N if and only if I is a real maximal ideal (cf. Proposition 3.3). We apply
this result and prove that for any countable CZ-family {Z1,Z2, ...,Zn, ...} of
Z[X], clβX(

⋃
i∈N Zi) =

⋃
i∈N clβXZi if and only if X is a pseudocompact space

(cf. Theorem 3.6).
In a general space, even a closed, countable union of zero-sets need not be

a zero-set; see 6P.5 in [8]. This was a motivation for introducing the class
of CZ-spaces in this paper. In section 4, we introduce CZ-spaces as those
spaces which in the closure of any countable union of zero-sets is a zero-set (cf.
Definition 4.1). We observe that a space X is a CZ-space if and only if the set
of basic z-ideals is closed under countable intersection, i.e., for every countable
subset f1,f2, ...,fn, ... of C(X) there exists f ∈ C(X) such that

⋂
i∈N Mfi = Mf

(cf. Lemma 4.3). Every open z-embedded subset of a CZ-space (hence open
C∗-embedded and cozero-sets) is a cz-space (cf. Proposition 4.4).

In section 5, we give some new equivalent conditions algebraic and topo-
logical for the class of cozero complemented spaces. It is proved that a space
X is cozero complemented if and only if the set of basic zo-ideals is closed
under countable intersection, i.e., for every countable subset f1,f2, ...,fn, ... of
C(X) there exists f ∈ C(X) such that

⋂
i∈N Pfi = Pf if and only if for each

f ∈ C(X) there exists a g ∈ C(X) such that Ann(f) = Pg (cf. Theorem 5.1).
A topological space X is called CAP-space if the closure of the interior of every
zero-set in X is a zero-set (cf. Definition 5.3). This class of spaces contains the
class of almost P-spaces and perfectly normal spaces. We conclude that every
CZ-space is a cozero complemented space and a CAP-space (cf. Proposition
5.6). Examples are given to show that the converse need not be true. We also
call a topological space X, CRZ (resp., CZ)-extended in a space T containing
it if for each regular-closed zero-set (resp., zero-set) Z ∈ Z[X], clT Z is a zero-
set in T (cf. Definition 5.7). Examples of CRZ (resp., CZ)-extended are given
(cf. Example 5.8). We prove that for a dense and Z#-embedded space X in
a space T , X is CAP and CRZ-extended in T if and only if T is CAP (cf.
Theorem 5.13). From this result, we get the following results (cf. Corollary
5.13):

(1) If X is a weakly Lindelöf dense space in a space T , then X is CAP and
CRZ-extended in T if and only if T is CAP.

(2) If X is CZ-extended in T , then X is CAP if and only if T is a CAP-
space.

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 2



Some class of topological spaces

(3) For any completely regular space X, βX is a CAP-space if and only if
X is a CAP and CRZ-extended in βX.

(4) If X is a CZ-extended in βX, then βX is a CAP-space if and only if
X is so.

2. Preliminaries

In this paper, all spaces are completely regular Hausdorff and C(X) (C∗(X))
is the ring of all (bounded) real-valued continuous functions on a space X.
For each f ∈ C(X), the zero-set of f denoted by Z(f) is the set of zeros of f
and cozf is the set X \Z(f) which is called the cozero-set of f. The set of all
zero-sets in X is denoted by Z[X] and for each ideal I in C(X), Z[I] is the set
of all zero-sets of the form Z(f), where f ∈ I. The support of f ∈ C(X), is
the set clX(X \Z(f)).
The space βX is known as the Stone-C̆ech compactification of X. It is char-
acterized as that compactification of X in which X is C∗-embedded as a dense
subspace. The space υX is the real compactification of X, if X is C-embedded
in this space as a dense subspace. For a completely regular Hausdorff space X,
we have X ⊆ υX ⊆ βX. Whenever Z = Z(f) ∈ Z[X], we denote Z(fβ) with
Zβ, where fβ is the extension of f to βX. By a z-ultrafilter on X is meant
a maximal z-filter, i.e., one not contained in any other z-filter. When M is a
real maximal ideal in C(X), we refer to Z[M] as a real-z-ultrafilter. Thus, the
real z-ultrafilters are those with the countable intersection property.
For any p ∈ βX, Op (resp., Mp) is the set of all f ∈ C(X) for which
p ∈ intβX clβX Z(f) (resp., p ∈ clβX Z(f)). Also, for A ⊆ βX, OA (resp., MA)
is the intersection of all Op(resp., Mp) where p ∈ A, and whenever A ⊆ X, we
denote it by OA (resp., MA).
The intersection of all minimal prime ideals of C(X) (resp., maximal ideals of
C(X)) containing f is denoted by Pf (resp., Mf ). It is proved in [5] that Pf =
{g ∈ C(X) : intX Z(f) ⊆ intX Z(g)} and Mf = {g ∈ C(X) : Z(f) ⊆ Z(g)}.
An ideal I of C(X) is a z-ideal if for each f ∈ C(X), Mf ⊆ I. For f ∈ C(X),
Ann(f) = {g ∈ C(X) : fg = 0} and it is easy to see that Ann(f) = MX\Z(f).
The reader is referred to [8] for more details on C(X).

3. Countable union of zeros-sets

Definition 3.1. A countable subfamily {Z1,Z2, ...,Zn, ...} of Z[X] is called a
CZ-family if

⋃∞
i=1 Zi is a zero-set.

If we consider the real numbers R with the usual topology and consider
a countable zero-set Z ∈ Z[R] (e.g., Z = Z(f), where f(x) = cos x), then
{{x} : x ∈ Z} is a CZ-family and in this space {{x} : x ∈ Q}, where Q is the
set of rational numbers, is not a CZ-family of Z[R].

Recall from [8], a space X is a P-space if C(X) is a regular space, i.e., every
zero-set in X is open. The next result shows that whenever X is a non P-space,
then there is a countable subfamily of Z[X] which is not a CZ-family.

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 3



F. Golrizkhatami and A. Taherifar

Proposition 3.2. The following statements are equivalent.

(1) Every countable subfamily of Z[X] is a CZ-family.
(2) X is a P -space.
(3) For every countable subset {Z1,Z2, ...,Zn, ...} of Z[X],

⋃
i∈N intX Zi is

a zero-set.
(4) Every z-ultrafilter on X is closed under countable union.

Proof. (1)⇒(2) As every cozero-set is a countable union of zeros-sets, this is
evident by [8, 4.J].

(2)⇒(3) By [8, 4.J], every zero-set is open, so this is obvious.
(3)⇒(4) Let Z[Mp] be a z-ultrafilter on X. By hypothesis and the fact that

every cozero-set is a countable union of the interior of zero-sets, every cozero-set
is a zero-set. That is every zero-set is a cozero-set. Now assume {Z(fi)|i ∈ N}
is a countable subset of Z[Mp]. Then for each i ∈ N, there exists a cozero-set
X\Z(gi) such that Z(fi) = X\Z(gi). Hence,

⋃
i∈N Z(fi) =

⋃
i∈N(X\Z(gi)) =

X\
⋂
i∈N Z(gi) = X\Z(g), for some g ∈ C(X). Again by hypothesis, X\Z(g)

is a zero-set, so
⋃
i∈N Z(fi) ∈ Z[M

p].
(4)⇒(1) Consider a countable subset S = {Z1,Z2, ...,Zn, ...} of Z[X]. For

each i ∈ N, define Z′i = Z1 ∪ Z2 ∪ ... ∪ Zi. Then S
′ = {Z′i : i ∈ N} has the

finite intersection property and
⋃
S =

⋃
S′. Now, consider the collection of all

zero-sets in X that contains finite intersections of the members of S′. This is
a proper z-filter in X. Thus this z-filter is contained in a unique z-ultrafilter,
say, Z[Mp] for some p ∈ βX. For each i ∈ N, we have Z′i ∈ Z[M

p]. Thus by
hypothesis,

⋃
i∈N Zi =

⋃
i∈N Z

′
i ∈ Z[M

p]. So S is a CZ-family. �

Proposition 3.3. Let I be an ideal of C(X). Then
⋃
i∈N Z(fi) ∈ Z[I] implies

fi ∈ I for some i ∈ N if and only if I is a real maximal ideal.

Proof. The sufficiency follows from [8, 5.15(a)].
Necessity. First, trivially I is a z-ideal. Next, for each n ∈ N, put Zn =

{x ∈ X : |f(x)| ≥ 1/n} and suppose that f /∈ I. We have Z(f) ∪ (
⋃
n∈N Zn) =

X ∈ Z[I]. Thus Zn ∈ Z[I], for some n ∈ N. But Zn is of the form Z(1 −gf)
for some g ∈ C(X), see Lemma 2.1 in [2]. Thus Z(1 − gf) ∈ Z[I] for some
g ∈ C(X). This shows that 1−gf ∈ I, i.e., I is a maximal ideal. Now we want
to prove I is a real ideal. By [8, Theorem 5.14], it is enough to show that I has
the countable intersection property. To see this, let for each n ∈ N, Zn ∈ Z[I]
and

⋂
n∈N Zn = ∅. Then

⋃
n∈N(X \ Zn) = X ∈ Z[I]. As every X \ Zn is a

countable union of zero-sets, we have a zero-set Z ∈ Z[I] contained in some
X \Zn. This contradicts with Zn ∈ Z[I]. �

It is well known that υX = {p ∈ βX : Mp is real }. Now by using the
above theorem we obtain the following result.

Corollary 3.4. Let p ∈ βX. Then p ∈ υX if and only if for each CZ-family
{Z(fi) : i ∈ N}, p ∈ clβX(

⋃∞
i=1 Z(fi)) implies p ∈ clβX Z(fi) for some i ∈ N.

We again apply Proposition 3.3 for proving the next results.

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 4



Some class of topological spaces

Corollary 3.5. Let {Z1,Z2, ...,Zn, ...} be a CZ-family in Z[X].
(1) clυX(

⋃∞
i=1 Zi) =

⋃∞
i=1 clυX Zi.

(2) The set {Zυ1 ,Zυ2 , ...,Zυn, ...,} is a CZ-family in Z[υX].

Proof. (1) Trivially,
⋃∞
i=1 clυX Zi ⊆ clυX(

⋃∞
i=1 Zi). Now for the proof of the

other inclusion, let p ∈ clυX(
⋃∞
i=1 Zi). Then this and hypothesis imply that⋃∞

i=1 Zi ∈ Z[M
p]. Hence there exists i ∈ N such that Zi ∈ Z[Mp], by Propo-

sition 3.3. This shows that p ∈ clυX Zi, and so p ∈
⋃∞
i=1 clυX Zi. So we are

done.
(2) Let

⋃∞
i=1 Zi = Z, where Z ∈ Z[X]. By (1),

⋃∞
i=1 Z

υ
i =

⋃∞
i=1 clυX Zi =

clυX(
⋃∞
i=1 Zi) = clυX Z = Z

υ. �

The CZ-family condition for {Z1,Z2, ...,Zn, ...} in Part 1 of the above result
is necessary. For, consider Q as a subspace of R with the usual topology. As
mentioned the set {{x} : x ∈ Q} is not a CZ-family and for each x ∈ Q, {x} is
a zero-set. However, clR

⋃
x∈Q{x} 6=

⋃
x∈Q{x}.

Theorem 3.6. The following statements hold.

(1) For every CZ-family {Z1,Z2, ...,Zn, ...} of Z[X], we have
clβX(

⋃∞
i=1 Zi) =

⋃∞
i=1 clβX Zi if and only if X is pseudocompact.

(2) For every countable subset {Z1,Z2, ...,Zn, ...} of Z[X], we have
intβX clβX(

⋃∞
i=1 Zi) =

⋃∞
i=1 intβX clβX Zi, if and only if X is finite.

(3) For every countable subset {Z1,Z2, ...,Zn, ...} of Z[X], we have
intX(

⋃∞
i=1 Zi) =

⋃∞
i=1 intX Zi if and only if X is a P -space.

Proof. (1) Necessity. Let p ∈ βX\υX. Then for each n ∈ N, there exists Zn ∈
Z[Mp] such that

⋂∞
n=1 Zn = ∅. Thus

⋃∞
n=1(X\Zn) = X. As each X\Zn is a

countable union of zero-sets say
⋃∞
m=1 Zmn, so we have

⋃∞
n=1(

⋃∞
m=1 Zmn) = X.

This shows that the set {Zmn : m,n ∈ N} is a CZ-family in Z[X]. So, by
hypothesis,

⋃∞
n=1(

⋃∞
m=1 clβX Zmn) = βX. Thus there exist m,n ∈ N such

that Zmn ⊆ X \ Zn and p ∈ clβX Zmn. This shows that Zmn ∈ Z[Mp], a
contradiction.
Sufficiency, we have βX = υX, so this follows from Corollary 3.5.

(2) Necessity. Suppose that p ∈ βX and
⋃∞
n=1 Zn ∈ Z[O

p]. Then

p ∈ intβX clβX(
⋃∞
i=1 Zi) =

⋃∞
i=1 intβX clβX Zi.

Thus there exists n ∈ N such that p ∈ intβX clβX Zn, i.e., Zn ∈ Z[Op]. By
Proposition 3.3, Op = Mp and p ∈ υX (i.e., βX = υX). Thus by [8, 7L], X is
a P-space and βX = υX implies X is pseudocompact, by [8, 8A.2]. Hence by
[8, 4K.2], X is finite.
Sufficiency, X is finite, so this is obvious.

(3) Necessity. The proof is similar to the (2).
Sufficiency, X is a P-space. Thus by Proposition 3.2,

⋃∞
i=1 Zi is a zero-set

in X and so p ∈ intX(
⋃∞
i=1 Zi) implies

⋃∞
i=1 Zi ∈ Z[Op] = Z[Mp]. Thus, by

Proposition 3.3, Zi ∈ Z[Op], i.e., p ∈ intX Zi, for some i ∈ N. The other
inclusion always holds, so we are done. �

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 5



F. Golrizkhatami and A. Taherifar

4. CZ-space

Definition 4.1. A topological space X is called a CZ-space if the closure of
any countable union of zero-sets is a zero-set.

Example 4.2.

(1) By Proposition 3.2, every P-space is a CZ-space.
(2) Every perfectly normal space (e.g., a metric space) is a CZ-space. So

the space of real numbers with usual topology is a CZ-space which is
not a P-space.

(3) In [8, 6P.5], N is a closed discrete subset of the space Λ = βR\(βN\N),
hence is a closed countable union of zero-sets. However it is not a zero-
set. Thus Λ is not a CZ-space.

(4) In [8, 4.N], S is a non-discrete P-space and hence this is a CZ-space.
However it contains a closed subset which is not a zero-set. So this is
an example of CZ which is not a perfectly normal space.

Let us call a z-ultrafilter F a CZ-ultrafilter if for each countable subset
{Z1,Z2, ...,Zn, ...} of F, clX(

⋃
i∈N Zi) ∈F.

Lemma 4.3. The following statements are equivalent.

(1) The space X is a CZ-space.
(2) For every countable subset {f1,f2, ...,fn, ...} of C(X) there exists f ∈

C(X) such that
⋂
i∈N Mfi = Mf .

(3) Every z-ultrafilter on X is a CZ-ultrafilter.

Proof. (1)⇒(2) Consider an arbitrary countable subset {f1,f2, ...,fn, ...} of
C(X). By hypothesis, there exists f ∈ C(X) such that clX(

⋃
i∈N Z(fi)) =

Z(f). Thus we have:

M⋃
i∈N Z(fi)

= MclX(
⋃

i∈N Z(fi))
= MZ(f) = Mf.

Trivially M⋃
i∈N Z(fi)

=
⋂
i∈N Mfi . So we are done.

(2)⇒(3) Let {Z(f1),Z(f2), ...Z(fn), ...} be a countable subset of a z-ultrafilter
Z[Mp]. By hypothesis,

⋂
i∈N Mfi = Mf , for some f ∈ C(X). This equal-

ity shows that M⋃
i∈N Z(fi)

= MZ(f). Hence clX(
⋃
i∈N Z(fi)) = Z(f). Thus

clX(
⋃
i∈N Z(fi)) ∈ Z[M

p].
(3)⇒(1) Consider a countable subset S = {Z1,Z2, ...,Zn, ...} of Z[X]. For

each i ∈ N, define Z′i = Z1 ∪ Z2 ∪ ... ∪ Zi. Then S
′ = {Z′i : i ∈ N} has the

finite intersection property and
⋃
S =

⋃
S′. Now, consider the collection of

all zero-sets in X that contains finite intersections of members of S′. This is
a proper z-filter in X. Thus this z-filter is contained in a unique z-ultrafilter,
say, Z[Mp] for some p ∈ βX. For each i ∈ N, we have Z′i ∈ Z[M

p]. Thus by
hypothesis, clX(

⋃
i∈N Zi) = clX(

⋃
i∈N Z

′
i) ∈ Z[M

p]. So X is a CZ-space. �

Proposition 3.2 shows that whenever X is a P-space, then every z-ultrafilter
is a CZ-ultrafilter. However, if X is a CZ-space which is not a P-space (e.g.,
R with usual topology), then there is a CZ-ultrafilter which is not closed under
countable union.

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 6



Some class of topological spaces

Recall from [6], a subset S of the topological space X is z-embedded if each
zero-set of S is the restriction to S of a zero-set of X. Now we will see that the
open z-embedded subsets inherit the CZ-property from the space.

Proposition 4.4. The following statements hold.

(1) Every open z-embedded subspace of a CZ-space is a CZ-space.
(2) Every cozero-set in a CZ-space is a CZ-space.
(3) Every open C∗-embedded (resp., C-embedded) subspace of a CZ-space

is a CZ-space.

Proof. (1) Let S be an open z-embedded subspace of X and {Zi : i ∈ N} be a
countable subset of Z[S]. By hypothesis, for each i ∈ N, there exists Z′i ∈ Z[X]
such that Z′i ∩ S = Zi. X is a CZ-space, so there exists Z

′ ∈ Z[X] with
clX(

⋃
i∈N Z

′
i) = Z

′. It is easy to see that

clS(
⋃
i∈N

Zi) = clX(
⋃
i∈N

Zi) ∩S = clX(
⋃
i∈N

Z′i ∩S) ∩S = clX(
⋃
i∈N

Z′i) ∩S = Z
′∩S.

This shows that S is a CZ-space.
(2) By [6, Proposition 1.1], every cozero-set in X is an open z-embedded.

So this follows from (1).
(3) Trivially every open C∗-embedded (resp., C-embedded) subspace is a

z-embedded set, so this follows from (1). �

A space X is an F-space (resp., F ′-space) if disjoint cozero subsets of X
are contained in disjoint zero sets (resp., if disjoint cozero subsets have disjoint
closures). As every cozero-set is a countable union of zero-sets, whenever X is
a CZ-space, the closure of every cozero subset is a zero-set. Thus we obtain
the following result.

Corollary 4.5. If X is an F ′-space and a CZ-space, then it is an F -space.

In the sequel we characterize some topological properties of the classes of
CZ-spaces. Recall from [10] that if f : X → Y is a continuous surjection map
and f(Z[X]) ⊆ Z[Y ], then f is said to be zero-set preserving. The following
result is Lemma 3.20 of [10].

Lemma 4.6. An open perfect surjection is zero-set preserving.

Theorem 4.7. The following statements hold.

(1) If f : X → Y is open and zero-set preserving and X is CZ, then Y is
CZ.

(2) If X is compact and X ×Y is CZ, then Y is CZ.

Proof. (1) Let {Z(fi) : i ∈ N} be a countable subset of Z[Y ]. Then {f−1(Z(fi)) :
i ∈ N} ⊆ Z[X]. Since, for each i ∈ N, f−1(Z(fi)) = Z(fiof) ∈ Z[X]. X is a
CZ-space, so there exists Z(g) ∈ Z[X] such that clX(

⋃
i∈N f

−1(Z(fi))) = Z(g).
We claim that clY (

⋃
i∈N Z(fi)) = f(Z(g)), which is a zero-set in Y , by hypoth-

esis. To see this, let y ∈ clY (
⋃
i∈N Z(fi)). We have y = f(x), for some x ∈ X.

It is enough to show that x ∈ Z(g), i.e., x ∈ clX(
⋃
i∈N f

−1(Z(fi))). Let U be an

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 7



F. Golrizkhatami and A. Taherifar

open set in X containing x. Then f(U) is open in Y and containing y and hence
f(U)∩(

⋃
i∈N Z(fi)) 6= ∅. Thus f(U)∩Z(fi) 6= ∅, for some i ∈ N. This implies

U ∩f−1(Z(fi)) 6= ∅, for some i ∈ N. Hence U ∩ (
⋃
i∈N f

−1(Z(fi))) 6= ∅, i.e.,
x ∈ clX(

⋃
i∈N f

−1(Z(fi))). Now assume y = f(x) ∈ f(Z(g)), where x ∈ Z(g)
and G be an open set in Y containing y. Then x ∈ f−1(G), which is open in
X. Thus f−1(G) ∩

⋃
i∈N f

−1(Z(fi))) 6= ∅. Hence f−1(G) ∩ f−1(Z(fi)) 6= ∅,
for some i ∈ N. This implies G∩Z(fi) 6= ∅, i.e., G∩ (

⋃
i∈N Z(fi)) 6= ∅. Thus

y ∈ clY (
⋃
i∈N Z(fi)).

(2) The map πY : X ×Y → Y is an open perfect map (since X is compact)
and surjective. Thus it is zero-set preserving, by Lemma 4.6. So this follows
from (1). �

As we found algebraic equivalent for a CZ-space in Lemma 4.3 and the fact
that C(X) ' C(υX) we obtain the following result.

Proposition 4.8. The following statements hold.

(1) If C(X) is isomorphic with C(Y ) (as two rings) and X is a CZ-space,
then Y is a CZ-space.

(2) X is a CZ-space if and only if υX is a CZ-space.
(3) If X is pseudocompact and CZ, then βX is a CZ-space.

5. CZ-space and other classes of topological spaces

A space X is cozero complemented if, given any cozero set U, there is a
disjoint cozero set V such that U∪V is dense in X. In [4], this class of space is
called m-space, i.e., every prime zo-ideal of C(X) is minimal. By Proposition
1.5 in [4], X is cozero complemented if and only if for every zero-set Z ∈ Z[X]
there exists a zero-set F ∈ Z[X] such that Z∪F = X and intX Z∩intX F = ∅.
By Corollary 5.5 in [9], this is equivalent to compactness of the space of minimal
prime ideals of C(X). In this section we give some another algebraic and
topological equivalent conditions for this class of spaces and conclude that
every CZ-space is a cozero complemented space. Some topological properties
of cozero complemented spaces are also characterized. We also introduce some
other classes of topological spaces which are used in the sequel. In particular,
we introduce a large class of topological spaces, which are called CAP-spaces
(cf. Definition 4.3), and observe that these spaces, although they are different
from the cozero-complemented spaces, behave in a similar manner as the latter
ones. We also provide several examples (cf. Examples 5.4 and 5.8).

Theorem 5.1. The following statements are equivalent.

(1) The closure of any countable union of the interior of zero-sets is the
closure of the interior of a zero-set.

(2) For every countable subset {f1,f2, ...,fn, ...} of C(X) there exists f ∈
C(X) such that

⋂
i∈N Pfi = Pf .

(3) For each countably generated ideal I of C(X), there exists g ∈ C(X)
such that Ann(I) = Pg.

(4) For each f ∈ C(X), there exists g ∈ C(X) such that Ann(f) = Pg.

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 8



Some class of topological spaces

(5) X is a cozero complemented space.
(6) Every support in X is the closure of the interior of a zero-set.

Proof. (1)⇒(2) let {f1,f2, ...,fn, ...} be a countable subset of C(X). Clearly,

O⋃
i∈N intX Z(fi)

=
⋂
i∈N

Pfi.

By hypothesis, clX(
⋃
i∈N intX Z(fi)) = clX(intX Z(f)), for some f ∈ C(X).

Since
⋃
i∈N intX Z(fi) is open, we have,

O⋃
i∈N intX Z(fi)

= M⋃
i∈N intX Z(fi)

= MclX(
⋃

i∈N intX Z(fi)
=

MclX(intX Z(f)) = MintX Z(f) = OintX Z(f) = Pf.

This implies
⋂
i∈N Pfi = Pf .

(2)⇒(3) Let I be an ideal of C(X) generated by {f1,f2, ...,fn, ...}. For fi
(1 ≤ i ≤ n) , there exists a countable subset {f1i,f2i, ...,fmi, ...} of C(X) such
that X \Z(fi) =

⋃
m∈N intXZ(fmi). Trivially we have,

Ann(I) = M⋃
i∈N(X\Z(fi)) = M

⋃
i∈N

⋃
m∈N

intXZ(fmi) =
⋂
i∈N

⋂
m∈N

OintXZ(fmi)

=
⋂
i∈N

⋂
m∈N

Pfmi.

By hypothesis,
⋂
i∈N

⋂
m∈N Pfmi = Pg for some g ∈ C(X). So we are done.

(3)⇒(4) Trivial.
(4)⇒(5) Let clX(X\Z(f)) be a support in X. By hypothesis, clX(intX Z(f)) =

clX(X \ Z(g)), for some g ∈ C(X). Get complement of two hands of the
equality, we have intX(clX(X \Z(f))) = intX(Z(g)). Hence clX(intX(clX(X \
Z(f))) = clX(intX(Z(g))). It is easy to see that clX(intX(clX(X \ Z(f))) =
clX(X \Z(f)). Thus clX(X \Z(f)) = clX(intX(Z(g))).

(5)⇒(1) Let {Z(f1),Z(f2), ...,Z(fn)} be a countable subset of Z[X]. By
hypothesis, for each i ∈ N, there exists gi ∈ C(X) such that intX Z(fi) =
intX clX(X \Z(gi)). Thus we have,

clX(

∞⋃
i=1

intX Z(fi) = clX(

∞⋃
i=1

intX clX(X \Z(gi)) = clX(
∞⋃
i=1

(X \Z(gi)))

= clX(X \
∞⋂
i=1

Z(gi).

There exists g ∈ C(X) such that
⋂∞
i=1 Z(gi) = Z(g). Thus clX(

⋃∞
i=1 intX Z(fi)) =

clX(X \Z(g), which is the closure of the interior of some zero-set, by hypoth-
esis. �

Proposition 5.2. The following statements are equivalent.

(1) Every support in X is a zero-set.
(2) The closure of any countable union of the interior of zero-sets is a

zero-set.

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 9



F. Golrizkhatami and A. Taherifar

(3) For every countable subset {f1,f2, ...,fn, ...} of C(X) there exists f ∈
C(X) such that

⋂
i∈N Pfi = Mf .

(4) For each countably generated ideal I of C(X), there exists g ∈ C(X)
such that Ann(I) = Mg.

(5) For each f ∈ C(X), there exists g ∈ C(X) such that Ann(f) = Mg.

Proof. (1)⇒(2) Let {Z(fi) : i ∈ N} be a countable subset of Z[X]. By
hypothesis, for each i ∈ N, there exists a cozero-set X \ Z(gi) such that
intX Z(fi) = X \Z(gi). Thus clX(

⋃∞
i=1 intX Z(fi)) = clX(

⋃∞
i=1(X \Z(gi))) =

clX(X \
⋂∞
i=1 Z(gi)). There exists g ∈ C(X) such that

⋂∞
i=1 Z(gi) = Z(g).

Hence clX(
⋃∞
i=1 intX Z(fi)) = clX(X \Z(g)), which is a zero-set, by hypothe-

sis.
(2)⇒(3) Let {f1,f2, ....,fn, ...} be a countable subset of C(X). Clearly,⋂∞
i=1 Pfi =

⋂∞
i=1 OintX Z(fi) = O

⋃∞
i=1

intX Z(fi). By hypothesis, clX(
⋃∞
i=1 intX Z(fi)) =

Z(f), for some f ∈ C(X). Hence O⋃∞
i=1

intX Z(fi) = M
⋃∞

i=1
intX Z(fi) = MZ(f) =

Mf , i.e.,
⋂∞
i=1 Pfi = Mf .

(3)⇒(4) The proof is similar to the proof of (2)⇒(3) of Theorem 5.1, step
by step.

(4)⇒(5) Trivial.
(5)⇒(1) Consider clX(X\Z(f)) as a support. By hypothesis, Ann(f) = Mg

for some g ∈ C(X). Thus MX\Z(f) = Mg = MZ(g). This implies clX(X \
Z(f)) = clX(Z(g)) = Z(g). �

A point p ∈ X is said to be an almost P-point if f ∈ Mp, intX Z(f) 6= ∅,
and X is called an almost P-space if every point of X is an almost P-point. It
is easy to see that a space X is an almost P-space if and only if every zero-set
in X is regular-closed. The reader is referred to [1], [7], [11] and [13], for more
details and properties of almost P -spaces.

Definition 5.3. A space X is called a C-almost P-space (briefly CAP-space)
if the closure of the interior of every zero-set in X is a zero-set.

Example 5.4.

(1) Clearly every almost P-space is a CAP-space.
(2) Every CZ-space is a CAP-space. For, let Z ∈ Z[X]. Then clX(X \

Z) = X \ intX Z = Z(f), for some f ∈ C(X). Thus clX intX(Z) =
clX(X\Z(f). As X is CZ, clX(X\Z(f)) is a zero-set, hence clX intX Z
is a zero-set. This implies every perfectly normal space (hence a metric
space) is a CAP-space. Thus R with usual topology is a CAP-space
which is not an almost P-space.

(3) Clearly every OZ-space (i.e., a space which in every regular-closed sub-
set is a zero-set) is a CAP-space.

Lemma 5.5. A space X is a CAP -space if and only if every basic zo-ideal of
C(X) is a basic z-ideal.

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 10



Some class of topological spaces

Proof. ⇒ Let Pf be a basic zo-ideal. By hypothesis, there exists g ∈ C(X)
such that clX(intX Z(f)) = Z(g). Thus we have,

Pf = OintX Z(f) = MintX Z(f) = MclX(intX(Z(f))) = MZ(g) = Mg.

⇐ Let Z(f) ∈ Z[X]. There exists g ∈ C(X) such that Pf = Mg. Since Pf =
OintX Z(f) and Mg = MZ(g). The equality Pf = Mg implies clX(intX Z(f)) =
Z(g). So X is a CAP-space. �

Proposition 5.6. The following statements hold.

(1) Every CZ-space is a cozero complemented space.
(2) Every support in X is a zero-set if and only if X is a cozero comple-

mented space and a CAP -space.

Proof. (1) X is a CZ-space. Thus every support is a zero-set, so by Proposition
5.2, for each f ∈ C(X) there exists g ∈ C(X) such that Ann(f) = Mg. As
Ann(f) is a zo-ideal, this implies Mg is a z

o-ideal and hence equals with Pg.
Now this follows from Theorem 5.1.

(2) First assume every support in X is a zero-set and Z ∈ Z[X]. Then
clX(X \Z) = Z(f) for some f ∈ C(X). This implies clX(intX(Z)) = clX(X \
Z(f)), i.e., X is a cozero complemented space, by Proposition 1.5 in [4]. Again
by hypothesis, clX(X\Z(f)) is a zero-set. Thus X is a CAP-space. Conversely,
let clX(X\Z) be a support. By hypothesis, clX(X\Z) = clX(intX(Z(g))), for
some g ∈ C(X). As X is a CAP-space, clX(intX(Z(g))) is a zero-set. Thus
clX(X \Z) is a zero-set. �

Part 2 of the above result shows that if X is a CAP-space which is not
a cozero complemented space or a cozero complemented space which is not
a CAP-space, then there is a non zero-set support in X. Thus X is not
a CZ-space. To see examples, first, consider the space X as the one point
compactification of an uncountable discrete space. Then X is an almost P-
space, since any non-empty Gδ of it contains an isolated point of the space,
hence is a CAP-space. But this is not a cozero complemented space and
hence is not a CZ-space, see example 3.3 in [4]. Next, consider the space
Λ = βR \ (βN \ N) in [8, 6P.5]. We have βΛ = βR. Thus βΛ is a cozero
complemented and hence Λ is a cozero complemented space. However, we
know that it is not a CZ. On the other hand, by [8, 6p.5], Λ is pseudocompact,
so if βΛ = υΛ is CZ, then we must have Λ is CZ, by Proposition 4.8, which
is not true. Thus βΛ = βR is not a CZ-space while we know that R is a
CZ-space.

Henriksen and Woods in [10] showed that for an uncountable discrete space

D, the Stone-C̆hech compactification βD of D is cozero complemented but
βD × βD is not, and so by part (1) of Proposition 5.6, βD × βD is not a
CZ-space.

Definition 5.7. A subspace X of a space T is called CRZ (resp., CZ)-
extended in T if for each regular-closed zero-set (resp., zero-set) Z ∈ Z[X],
clT Z is a zero-set in T .

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 11



F. Golrizkhatami and A. Taherifar

Example 5.8.

(1) Let X be a C∗-embedded in T . If X is C-embedded in T , then it is
CZ-extended in T . For, clT Z(f) = Z(f

o), for f ∈ C(X) and fo is the
continuous extension of f to T.

(2) Every pseudocomact space is a CZ-extended in βX. For, if X is pseu-
docompact, then βX = υX and for each Z ∈ Z[X], clβX Z = clυX Z =
Zυ = Zβ.

(3) Consider the infinite P-space X (e.g., the discrete space N). Then every
zero-set Z ∈ Z[X] is open and hence clβX Z is clopen in βX. Thus it is
a zero-set in βX. This says that X is a CZ-extended in βX. However
X need not be C-embedded in βX.

(4) Trivially every CZ-extended in a space containing it, is a CRZ-extended.
But the space Σ = N ∪{σ} in [8, 4M] is CRZ-extended in βΣ = βN
which is not a CZ-extended in βN, see [8, 6E].

(5) Trivially every CZ-extended X in a space containing it, is Z-embedded.
However a Z-embedded need not be a CZ-extended. For example, Σ
is Z-embedded in βΣ, but is not CZ-extended in it.

Recall from [10], let X and T be two completely regular spaces and X be
a subspace of T . X is said to be Z#-embedded in T if for each f ∈ C(X),
there exists a g ∈ C(T) such that clX(intX(Z(f))) = clT (intT (Z(g)))∩X. The
following lemma is proved in [10].

Lemma 5.9. If X is a subspace of T that is either open or dense, then the
following are equivalent.

(1) X is Z#-embedded in T .
(2) If Z ∈ Z[X], then there is a Zt ∈ Z[T ] such that intX Z = (intT Zt) ∩

X.

Lemma 5.10. Suppose that X is dense or open as well as being Z#-embedded
in a space T .

(1) If T is CAP , then so is X.
(2) If every support in T is a zero-set, then every support in X is a zero-set.

Proof. (1) First assume X is dense in T. We show X is a CAP-space. Let
Z ∈ Z[X]. By hypothesis and Lemma 5.9, there is Zt ∈ Z[T ] such that
intX Z = intT Z

t ∩X. Thus clX(intX Z) = clX(intT Zt ∩X) = clT (intT Zt ∩
X)∩X = clT (intT Zt)∩X. By hypothesis, there exists Z(f) ∈ Z[T] such that
clT (intT Z

t) = Z(f). Hence clX(intX Z) = Z(f) ∩ X which is a zero-set in
X. If X is open in T , then for each p ∈ clT (intT Zt) ∩ X and U open in X
containing p, we have U is open in T and so U∩intX Z = U∩intT (Zt)∩X 6= ∅.
Hence clX(intX Z) = clT (intT Z

t) ∩X. So we are done.
(2) Every support in T is a zero-set, so by Proposition 5.6, T is a cozero

complemented space and a CAP-space. Thus X is a cozero complemented
space, by Lemma 2.5 in [10]. Also, by part 1, X is CAP. Now Proposition 5.6
implies every support in X is a zero-set. �

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 12



Some class of topological spaces

Since every open (dense) z-embedded subset is Z#-embedded, and every
cozero (resp., C∗-embedded)-subset is z-embedded in any space containing it,
the above lemma implies next result.

Corollary 5.11. The following statements hold.

(1) Every open (dense) z-embedded subspace of a CAP -space is CAP .
(2) Every cozero-set in a CAP -space is CAP .
(3) Every open (dense) C∗-embedded in a CAP -space is a CAP .

Recall that if every open cover of a space X contains a countable subfamily
whose union is dense in X, then X is called weakly Lindelöf space. Every Lin-
delöf space and every ccc-space is weakly Lindelöf space, while an uncountable
discrete space is not a weakly Lindelöf space.

Corollary 5.12. If S ∩ W is a weakly Lindelöf space where S is a dense
subspace and W is an open subspace of a CAP -space T , then S ∩W is CAP .

Proof. Similar to the proof of Theorem 2.6 in [10], we have clW (S ∩ W) =
clT (S ∩ W) ∩ W = clT W ∩ W = W . Thus S ∩ W is dense in W . So, since
S ∩ W is weakly Lindelöf, so is W . We have W is open in T, hence this is
Z#-embedded, by Lemma 2.4 of [10]. Thus W is CAP, by Lemma 5.10. But
S ∩W is dense in W , so by Lemma 2.4 in [10], S ∩W is Z#-embedded in W .
Thus by Lemma 5.10, we are done. �

Theorem 5.13. Let X be dense and Z#-embedded in a space T . Then the
following statements hold.

(1) T is a CAP -space if and only if X is a CAP -space and CRZ-extended
in T .

(2) Every support in T is a zero-set if and only if every support in X is a
zero-set and X is CRZ-extended in T .

Proof. (1)⇒ Part 1 of Lemma 5.10 shows X is a CAP-space. Now, assume
Z(f) ∈ Z[X] be a regular-closed zero-set. Then Z(f) = clX(intX(Z(f))) and
by hypothesis, there exists ft ∈ C(T) such that intX Z(f) = intT Z(ft) ∩ X.
T is CAP, hence there is a g ∈ C(T) such that clT intT Z(ft) = Z(g). Thus
we have,

clT Z(f) = clT (clX(intX(Z(f)))) = clT (clX(intT (Z(f
t)) ∩X))

= clT (intT (Z(f
t)) ∩X) = clT (intT (Z(ft))) = Z(g).

This completes the proof.
⇐ Let Z(ft) ∈ Z[T]. Then Z(ft) ∩ X ∈ Z[X]. X is a CAP-space, hence

clX(intX(Z(f
t)∩X)) = Z(g) for some g ∈ C(X). Thus Z(g) is a regular-closed

set. On the other hand, we have intX(Z(f
t) ∩X) = intT Z(ft) ∩X. To see it,

intT Z(f
t) ∩X is open in X and contained in Z(ft) ∩X. Thus it is contained

in intX(Z(f
t) ∩ X). Now, suppose p ∈ intX(Z(ft) ∩ X). Then p ∈ X and

there is an open subset U of T such that p ∈ U ∩ X ⊆ Z(ft). This implies

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 13



F. Golrizkhatami and A. Taherifar

p ∈ U ⊆ clT U = clT (U ∩X) ⊆ Z(ft), i.e., p ∈ intT Z(ft)∩X. It is easy to see
that;

clT (intT (Z(f
t))) = clT (intT (Z(f

t)) ∩X) = clT (intX(Z(ft) ∩X)) =
clT (clX(intX(Z(f

t) ∩X)) = clT (Z(g)).
As X is CRZ-extended in T , clT Z(g) is a zero-set in T, so clT (intT (Z(f

t))) is
a zero-set in T , i.e., T is a CAP-space.

(2) ⇒ Part 2 of Lemma 5.10 implies every support in X is a zero-set. On the
other hand, T is CAP, by Proposition 5.6. Thus by Part (1), for any regular-
closed zero-set Z in X, clT Z is a zero-set in T , i.e., X is CRZ-extended in
T .
⇐ Every support in X is a zero-set, hence X is a cozero complemented space

and a CAP-space, by Proposition 5.6. This implies T is a cozero complemented
space, by Theorem 2.8 in [10]. Moreover, T is a CAP, by part (1). Now, again
by using Proposition 5.6, we have every support in T is a zero-set. �

It is easy to see that if X is CZ-extended in T, then X is Z#-embedded in
T . So we conclude the following result from the above theorem.

Corollary 5.14. The following statements hold.

(1) If X is weakly Lindelöf and dense in T , then T is CAP if and only if
X is CAP and CRZ-extended in T .

(2) If X is CZ-extended in T , then X is CAP if and only if T is a
CAP -space.

(3) βX is a CAP -space if and only if X is a CAP and CRZ-extended in
βX.

(4) If X is a CZ-extended in βX, then βX is a CAP -space if and only if
X is so.

(5) Every support in βX is a zero-set if and only if every support in X is
a zero-set and X is CRZ-extended in βX.

(6) If X is a CZ-extended in βX, then every support in βX is a zero-set
if and only if every support in X is a zero-set.

6. Directions and some questions

CZ − space. As we have shown a space X is CZ if and only if the set of
basic z-ideals is closed under countable intersection. This shows that this class
of spaces is important. So we may have focus on the spaces Max (C(X)) and
Spec(C(X)), with Zariski topology, whenever X is a CZ-space. On the other
hand, since the set L = {Mf : f ∈ C(X)} is a lattice with two operations:
Mf ∨ Mg = Mf2+g2 and Mf ∧ Mg = Mf ∩ Mg = Mfg. So, we have X is a
CZ-space if and only if for every countable subset S of L, ∧S ∈ L. This can
help us to investigate more properties of CZ-spaces by the lattice properties of
L.
We have seen that if X is a pseudocompact and CZ, then βX is a CZ-space.
As an example, we have seen that R is a CZ-space, but βR is not a CZ.
However, this is a remainder question as follows:

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 14



Some class of topological spaces

Question 6.1. When is βX a CZ-space?

We also have some other questions as follows:

Question 6.1. Is every CZ hereditarily CZ (i.e., all subspaces are CZ)?

Question 6.2. Is the product of two (infinite) CZ-spaces a CZ?

Question 6.3. Suppose X ×Y is a CZ-space. Must X or Y be CZ?

CAP − space. We observed that the class of CAP-spaces behaves like
cozero-complemented spaces. So we may investigate other properties of CAP-
spaces. It also is important to know the relation between CAP and cozero-
complemented spaces. We know that X is cozero-complemented if and only if
Min (C(X)) is compact. So this is a motivation to ask the following questions:

Question 6.4. What is Min (C(X)), when X is a CZ?

Question 6.5. What is Max (C(X)), when X is a CZ?

We also have some questions as follows:

Question 6.6. Characterize the CAP spaces which are hereditarily CAP.

Question 6.7. Is the product of two (infinite) CAP-spaces a CAP?

Acknowledgements. The authors would like to thank the referee for her/his
thorough reading of this paper and her/his comments which led to a much
improved paper.

References

[1] F. Azarpanah, On almost P-spaces, Far East J. Math. Sci. 1 (2000), 121–132.
[2] F. Azarpanah, M. Ghirati and A. Taherifar, Closed ideals in C(X) with different repere-

sentations, Houston Journal of Mathematics 44, no. 1 (2018), 363–383.

[3] F. Azarpanah, A. A. Hesari, A. R. Salehi and A. Taherifar, A Lindelöfication, Topology
Appl. 245 (2018), l46–61.

[4] F. Azarpanah and M. Karavan, On nonregular ideals and zo-ideals in C(X), Czechoslo-

vak Math. J. 55 (2005), 397–407.
[5] F. Azarpanah, O. A. S. Karamzadeh and A. Rezai Aliabad, On z-ideals in C(X), Fund.

Math. 160 (1999), 15–25.
[6] R. L. Blair and A. W. Hager, Extension of zero-sets and real-valued functions, Math. Z.

136 (1974), 41–52.
[7] F. Dashiel, A. Hager and M. Henriksen, Order-Cauchy completions and vector lattices

of continuous functions, Canad. J. Math. XXXII (1980), 657-685.

[8] L. Gillman and M. Jerison, Rings of Continuous Functions, Springer, 1976.

[9] M. Henriksen and M. Jerison, The space of minimal prime ideals of a commutative ring,
Trans. Amer. Math. Soc. 115 (1965), 110–130.

[10] M. Henriksen and G. Woods, Cozero complemented spaces; when the space of minimal
prime ideals of a C(X) is compact, Topology Appl. 141 (2004), 147–170.

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 15



F. Golrizkhatami and A. Taherifar

[11] R. Levy, Almost P-spaces, Canad. J. Math. 2 (1977), 284–288.

[12] A. Taherifar, Some new classes of topological spaces and annihilator ideals, Topology

Appl. 165 (2014), 84–97.
[13] A. I. Veksler, P ′-points, P ′-sets, P ′-spaces. A new class of order-continuous measures

and functionals, Sov. Math. Dokl. 14 (1973), 1445–1450.

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 16