@ Appl. Gen. Topol. 22, no. 1 (2021), 79-89doi:10.4995/agt.2021.13608
© AGT, UPV, 2021

Ideal spaces

Biswajit Mitra and Debojyoti Chowdhury

Department of Mathematics, University of Burdwan, Burdwan 713104, West Bengal, India.

(bmitra@math.buruniv.ac.in, sankha.sxc@gmail.com)

Communicated by A. Tamariz-Mascarúa

Abstract

Let C∞(X) denote the family of real-valued continuous functions which
vanish at infinity in the sense that {x ∈ X : |f(x)| ≥ 1

n
} is compact in

X for all n ∈ N. It is not in general true that C∞(X) is an ideal of
C(X). We define those spaces X to be ideal space where C∞(X) is an
ideal of C(X). We have proved that nearly pseudocompact spaces are
ideal spaces. For the converse, we introduced a property called “RCC”

property and showed that an ideal space X is nearly pseudocompact

if and only if X satisfies ”RCC” property. We further discussed some

topological properties of ideal spaces.

2010 MSC: 54F65; 54G20; 54D45; 54D60; 54D99.

Keywords: rings of continuous functions; CK(X) and C∞(X); nearly pseu-
docompact spaces; RCC properties.

1. Introduction

In this paper, by a space we shall mean completely regular Hausdorff space,
unless otherwise mentioned. As usual C(X) and C∗(X) are real-valued con-
tinuous and bounded continuous functions respectively. They are commutative
rings with 1 under usual pointwise addition and multiplication. Rigorous and
systematic developments of these two rings are made in the classic monograph
of L. Gillman and M. Jerrison entitled “Rings of Continuous Functions”[7].
In fact most of the symbols, definitions and results are hired from the above
book. There are two important classes of subrings of C(X), namely, C∞(X)
and CK(X), where C∞(X) is the family of all those continuous functions such

Received 28 April 2020 – Accepted 29 October 2020

http://dx.doi.org/10.4995/agt.2021.13608


B. Mitra and D. Chowdhury

that {x : |f(x)| ≥ 1
n
} is compact for all n ∈ N and CK(X) is the family of all

functions f ∈ C(X) whose support, that is clX(X\Z(f)), is compact, where
Z(f) = {x ∈ X : f(x) = 0}. Both the subrings are in fact subrings of C∗(X)
also. Even more they are ideal in C∗(X). But though CK(X) is an ideal of
C(X) , C∞(X) need not be an ideal of C(X). Immediate example can be
cited if we count X = N. In this paper we have worked on those spaces X
for which C∞(X) is an ideal of C(X). For the sake of convenience, we define
these spaces as ideal space. In this context it ought to be relevant to mention
that Azarpanah et. al in [14], [2], already did some works in this area. They
have shown that every pseudocompact space is an ideal space and every ideal
space is pseudocompact if the space is locally compact. They also introduced
∞-compact space, the space where CK(X) = C∞(X), which is trivially an
ideal space.

In this paper we have shown that every nearly pseudocompact space, intro-
duced by Henriksen and Rayburn in [10], is also ideal space. We have further
introduced a criteria, so called RCC property, a generalization of locally com-
pact property, to go for converse. In fact we have shown that a nearly pseu-
docompact space is ideal space and an ideal space is nearly pseudocompact if
and only if the space satisfies RCC property. Throughout the paper we have
given many examples and counter examples. Henriksen and Rayburn in their
paper [10] have shown that every anti-locally realcompact space, i.e., having no
point with realcompact neighbourhood, is a nearly pseudocompact space. All
the examples of nearly pseudocompact spaces that they cited, are anti-locally
realcompact but they did not produce any example of a nearly pseudocompact
space which is not anti-locally realcompact. Here we have cited such example
[Example 4.14]. At the end we tried to explore few topological properties of
ideal spaces and finally have shown that if Xand Y are nearly pseudocompact,
then X × Y is nearly pseudocompact if and only if X × Y is ideal space.

2. Preliminaries

As we already mentioned that most of the basic symbols and terminologies
followed the book, The Rings of Continuous Functions, by L. Gillman and M.
Jerrison [7], yet for ready references, we include few basic notations, definitions
and related results that will be repeatedly used here. For each f ∈ C(X),
Z(f) = {x ∈ X : f(x) = 0} is called the zero set of f. The complement of zero
set is called cozero set, denoted as coz f. For each space X, βX is the largest
compactification of X where every compact-valued continuous function can be
continuously extended, referred as Stone-Čech compactification of X. It is also
the largest compactification of X in which X is C∗-embedded. Similarly υX is
the largest realcompact subspace of βX in which X is C-embedded. A space
is realcompact if it can be embedded as a closed subspace in the product of
reals. The υX is referred as Hewitt-Nachbin completion of X or simply Hewitt
realcompactification of X. Compactness and realcompactness can be easily
characterized respectively by showing X = βX and X = υX. A space X is

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Ideal spaces

pseudocompact (i.e, every realvalued continuous function on X is bounded) if
and only if βX = υX or equivalently βX\X = υX\X. Both realcompactness
and pseudocompactness are significant generalizations of compactness. How-
ever the realcompact and pseudocompact jointly enforce compact. A subset S
of X is relatively pseudocompact if every continuous function on X is bounded
over S. R.L. Blair and M.A. Swardson proved that a subset S of a space X is
relatively pseudocompact if clβXS ⊆ υX [3, Proposition 2.6]. Henriksen and
Rayburn in their paper [10] defined a space X to be nearly pseudocompact if
υX\X is dense in βX\X, a generalization of pseudocompact space. They have
proved the following characterization of nearly pseudocompact spaces.

Theorem 2.1. A space X is nearly pseudocompact if and only if X = X1 ∪X2,
where X1 is a regular closed almost locally compact pseudocompact subset,
and X2 is a regular closed anti-locally realcompact subset and int(X1∩X2) =
∅

In the year 1976, Rayburn in his paper [13] defined hard set in X. A subset
H of X is hard in X if H is closed in X ∪ K where K = clβX(υX\X). It is
clear that every hard set in X is closed in X. However he has also provided a
characterization of hardness of a closed subset in this paper as follows.

Theorem 2.2. A closed subset F of X is hard in X if and only if there exists
a compact set K such that for any open neighbourhood V of K, there exists a
realcompact subset P of X so that F\V and X\P can be completely separated
in X.

In particular H is hard if it is completely separated from the complement of
a realcompact subset of X.

Henriksen and Rayburn in [10] described few more characterizations as fol-
lows.

Theorem 2.3. A space X is nearly pseudocompact if and only if every hard
set is compact if and only if every regular hard set (i.e, a regular closed set
which is hard in X) is compact.

They have further proved the following theorem which is relevant in this
paper.

Theorem 2.4. Every regular closed subset of a nearly pseudocompact space is
a nearly pseudocompact space.

In the year 2005, Mitra and Acharyya in their paper [12] introduced two sub-
rings of C(X), CH(X) and H∞(X), analogical to the rings CK(X) and C∞(X).
As per their definition, CH(X) = {f ∈ C(X) : clX(X\Z(f)) is hard in X} and
H∞(X) := {f ∈ C(X) : {x ∈ X : |f(x)| ≧

1
n
}is hard in X}. They have

shown that CH(X) is an ideal of C(X). But no conclusion was made regard-
ing H∞(X). However they have given the following characterization of nearly
pseudocompact spaces using CH(X) and H∞(X).

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B. Mitra and D. Chowdhury

Theorem 2.5. The following statements are equivalent for a space X:

(1) X is nearly pseudocompact.
(2) CK(X) = CH(X).
(3) C∞(X) = H∞(X)
(4) CH(X) ⊆ C

∗(X)
(5) H∞(X) ⊆ C

∗(X)
(6) H∞(X)

⋂

C∗(X) = C∞(X)
(7) CH(X)

⋂

C∗(X) = CK(X)

3. Ideal spaces-a generalization of nearly pseudocompact space

In this section we shall formally study ideal spaces. We begin with the
definition of ideal spaces that we already introduced above.

Definition 3.1. A space X is called an ideal space if C∞(X) is an ideal of
C(X).

Azarpanah and Soundarajan in [2], gave a nice characterization of ideal
space.

Theorem 3.2. A space X is an ideal space if and only if every locally compact
σ-compact subset is relatively pseudocompact if and only if every open locally
compact subset is relatively pseudocompact.

They further proved that within the class of local compact spaces, the no-
tions of ideal space and pseudocompact space are identical.

However here use of locally compact condition does not require to prove
that every pseudocompact space is ideal. On the other hand, it will be shown
here that, ideal space generalizes nearly pseudocompact and ∞-compact space.
That the ∞-compact space is ideal trivially follows from its definition, hence
we shall mainly concentrate on nearly pseudocompact space. Before showing
that, we shall first show that H∞(X) is indeed an ideal of C(X) which was
unanswered in the paper of Mitra and Acharyya [12]. In fact we shall show that
H∞(X) = CRC(X), where CRC(X) = {f ∈ C(X) : coz f is realcompact} and
that CRC(X) is an ideal of C(X), follows from the fact that realcompact co-
zero sets are closed under finite, in fact countable union and realcompactness
is a co-zero hereditary property. The notion of CRC(X) was introduced by
T. Isiwata in [11]. Further, Azarpanah et.al in [[1], Theorem 3.8] produced
another characterization of this ideal. Here we established the following result.

Theorem 3.3. H∞(X) = CRC(X).

Proof. We know that X\Z(f) = ∪n[x ∈ X : |f(x)| 	
1
n
], for f ∈ C(X).

If f ∈ H∞(X), [x ∈ X : |f(x)| ≥
1
n
] is hard in X and hence realcompact.

Since [x ∈ X : |f(x)| 	 1
n
] is a co-zero subset of [x ∈ X : |f(x)| ≥ 1

n
],

[x ∈ X : |f(x)| 	 1
n
] is also realcompact. As countable union of realcompact

co-zero set is realcompact, so X\Z(f) is also realcompact. So f ∈ CRC(X).

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Ideal spaces

Conversely, if f ∈ CRC(X), then the zero set [x ∈ X : |f(x)| ≥
1
n
] is completely

separated from the complement of the realcompact co-zero set X\Z(f), for all
n. Hence the [x ∈ X : |f(x)| ≥ 1

n
] is hard in X, for all n. Thus f ∈ H∞(X). �

Corollary 3.4. Every nearly pseudocompact space is an ideal space.

Proof. Since a nearly pseudocompact space, H∞(X) = C∞(X) [Theorem 2.5]
and H∞(X) is always an ideal of C(X), C∞(X) is also an ideal of C(X). �

However the following example shows that the converse is not true.

Example 3.5. Let X1 be a non-compact realcompact space where the clo-
sure of the set D of points having compact neighbourhood is compact, e.g.
[(−∞, 0) ∩ Q] ∪ [0, 1] ∪ [(1, ∞) ∩ Q]. Then X1 × [0, ω1], where [0, ω1] is the
space of all ordinals less than or equal to the first uncountable ordinal ω1, is
an ideal space which is not nearly pseudocompact. The reason is quite simple.
If we take U, a locally compact, σ-compact space, then U ⊆ D × ω1. Now
clD × ω1, being pseudocompact space, U is relatively pseudocompact; hence
an ideal space. But it is not nearly pseudocompact, as X1 ×{0} being a regular
closed subset of X1 × [0, ω1], not nearly pseudocompact as it is non-compact
real-compact.

In the above example, U is actually contained in a compact space as the right
projection of U into [0, ω1] is also locally compact and σ-compact and hence is
bounded by a compact subset K. Thus U is then contained in clD × K.

This type of space is called ∞-compact space.

Definition 3.6. A space is called ∞-compact if CK(X) = C∞(X).

From [2, Proposition 2.1], we know the following result.

Theorem 3.7. A space is ∞-compact if and only if every open locally compact
σ-compact subset is bounded by a compact set.

This tells us that every ∞-compact space is an ideal space. The following
example is an ideal space which is not ∞-compact.

Example 3.8. As in Example 3.5, we take X1 making product with Tychonoff
plank T . That it is an ideal space but not nearly pseudocompact follows along
the same lines of argument as in example 3.5. We show here that it is not
even ∞-compact. U = ∪n(1/4, 1/2) × ([0, ω1] × {n}) is open locally compact
σ-compact subset. But U is not contained in any compact set as then, it’s
projection into Tychonoff plank would be covered by a compact set. But the
right edge, the copy of N, being a closed subset of U would be compact, which
is not true.

We therefore have the following parallel strictly forward implications.

compact →֒ pseudocompact →֒ nearly pseudocompact →֒ ideal space.
compact →֒ ∞ − compact →֒ ideal space.

Next we give an example of a pseudocompact space which is not ∞-compact.

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B. Mitra and D. Chowdhury

Example 3.9. We take Tychonoff plank [0, ω1]×[0, ω0]\{(ω1, ω0)}. The union
of the horizontal line [0, ω1] × {n}, n ∈ N is open, locally compact, σ-compact
subset but not bounded by any compact set.

4. Introduction of RCC property and its significance

We start first with the following definition;

Definition 4.1. A space X is said to satisfy RCC property if the set of
points having compact neighbourhood and the set of points having realcompact
neighbourhood are same.

As for instance, locally compact spaces satisfy RCC property.

Theorem 4.2. Every nearly pseudocompact space satisfies RCC property

Proof. As clβX(υX\X) = clβX(βX\X), clβX(υX\X)∩X = clβX(βX\X)∩X.
But clβX(υX\X) ∩ X and clβX(βX\X) ∩ X are respectively the set of points
in X having no realcompact and compact neighbourhoods. Hence the set of
points having compact neighbourhood is identical with that of having realcom-
pact neighbourhood. Hence every nearly pseudocompact space satisfies RCC
property. �

In 2004, Aliabad et. al [14, Corollary 1.2] proved that for a locally compact
Hausdorff space X, C∞(X) is ideal of C(X) if and only if X is pseudocom-
pact space. In the year 1980, Henriksen and Rayburn, [10, Theorem 3.9],
proved that regular closed almost locally compact subset of nearly pseudocom-
pact space is pseudocompact which in turn implies that locally compact nearly
pseudocompact space is pseudocompact. So we have the following theorem.

Theorem 4.3. Under the assumption of locally compactness, ideal, nearly
pseudocompact and pseudocompact spaces are identical.

In the next theorem we shall show that under RCC condition, nearly
pseudocompact and ideal spaces are same.

In [14], Aliabad et. al introduced an ideal Clσ(X) := {f ∈ C(X) : coz fis
locally compact and σ − compact}. Clσ(X) is a z − ideal of C(X) and by the
result of [14, Proposition 3.2], Clσ(X) is the smallest z-ideal of C(X) containing
C∞(X). Further we note that Clσ(X) ⊆ CRC(X).

Theorem 4.4. A space X is nearly pseudocompact if and only if X satisfies
RCC property and Clσ(X) ⊂ C

∗(X)

Proof. Let X be nearly pseudocompact. Then X satisfies RCC property by
theorem 4.2 . Furthermore H∞(X) ⊆ C

∗(X), by theorem 2.5 (5). By theorem
3.3, H∞(X) = CRC(X). Thus Clσ(X) ⊆ CRC(X) = H∞(X) ⊆ C

∗(X).

Conversely, suppose X is not nearly pseudocompact space. Since X satisfies
RCC property, then the set DX of points with compact neighbourhood is
non-empty, otherwise X would be anti-locally realcompact and hence nearly

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Ideal spaces

pseudocompact [10, Corollary 3.5]. Now for all f ∈ H∞(X), each point of
X\Z(f) has a realcompact neighbourhood [Theorem 3.3]. As the space sat-
isfies RCC property, ∀f ∈ H∞(X) we have X\Z(f) ⊆ DX. As X is not
nearly pseudocompact, by theorem 2.5(5), there exists f ∈ H∞(X) such that
f is unbounded on X\Z(f). Thus there exists a copy N of N in X\Z(f),
C-embedded in X, [7, corollary 1.20]. We consider a continuous function h on
X so that h(n) 	 n3, for all n ∈ N.

As N is C-embedded in X, N, hence any of its subset, is closed in X. So
clβXN\N is contained in βX\X. So clβX(βX\X) ∪ {m ∈ N : m 6= n} is
indeed a closed subsets of βX for each n ∈ N. Due to complete regularity, for
each n , there exists a continuous function ĝn : βX → R, such that

ĝn(x) =

{

= 0 when x ∈ clβX(βX\X) or x ∈ {m : m 6= n}.

= n
3

h(n)
when x = n

Without loss of generality, we assume |ĝn| ≤ 1 as for each n,
n
3

h(n)
� 1.

We take as usual, ĝ =
∑

n
gn
n2

. Then ĝ ∈ C(βX). Let g := ĝ|
X
. Clearly

ĝ is the Stone-extension of g. As ĝ vanishes everywhere on βX\X, closure of
{x ∈ X : |g(x)| ≥ 1

n
} in βX must not intersect in βX\X. Hence {x ∈ X :

|g(x)| ≥ 1
n
} = clβX{x ∈ X : |g(x)| ≥

1
n
} and is therefore compact. Thus

g ∈ C∞(X).
Now as g ∈ C∞(X), g ∈ Clσ(X). Then hg ∈ Clσ(X). Moreover hg(n) =

n, ∀n and hence is unbounded. �

Corollary 4.5. A space is nearly pseudocompact if and only if it is an ideal
space satisfying RCC property.

Proof. Every nearly pseudocompact space is ideal [Theorem 3.4] and satisfies
RCC property [Theorem 4.2]. Conversely suppose X is an ideal space hav-
ing RCC property, then let f ∈ Clσ(X). Then X\Z(f) is locally compact
and σ-compact and hence relatively pseudocompact by theorem 3.2 and hence
Clσ(X) ⊂ C

∗(X). By the above theorem, X is nearly pseudocompact. �

Remark 4.6. Although it is evident from Corollary 4.5 that we can not drop
the condition RCC. However in support of the above corollary, we do refer the
space given in example 3.5 that does not satisfy RCC property. But it is an
ideal space which is not nearly pseudocompact. (0, 1)×[0, ω1] is precisely the set
of points which have compact neighbourhood. But each point of X1×[0, ω1] has
realcompact neighbourhood X1 is realcompact and [0, ω1] is locally compact.
Hence the space does not satisfy RCC .

In the year 2001, Azarpanah and Soundararajan [ Proposition 2.4, [2]],
proved the following result.

Theorem 4.7 (Proposition 2.4, [2]). For any space X, let Cψ(X) be the family
of all real-valued continuous functions over X with pseudocompact support.

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B. Mitra and D. Chowdhury

Then C∞(X) is subset of Cψ(X) if and only if every open locally compact
subset of X is relatively pseudocompact.

In the year 2005, Henriksen and Mitra proved the following lemma.

Theorem 4.8 (Lemma 2.10, [9]). A function f ∈ C(X) is in Cψ(X) if and
only if fg ∈ C∗(X) whenever g ∈ C(X).

The following lemma directly follows as a corollary from theorems 3.2, 4.7
and 4.8 above.

Lemma 4.9. A space X is ideal if and only if ∀f ∈ C(X) and ∀g ∈ C∞(X),
fg ∈ C∗(X).

Proof. Suppose X is ideal. By theorem 3.2, every open locally compact subset
of X is relatively pseudocompact. By theorem 4.7, C∞(X) ⊆ Cψ(X). Hence by
theorem 4.8, for all f ∈ C(X) and for all g ∈ C∞(X), fg ∈ C

∗(X). Conversely
suppose ∀g ∈ C∞(X), fg ∈ C

∗(X). By theorem 4.8, we conclude that g ∈
Cψ(X). So C∞(X) ⊆ Cψ(X). By theorem 4.7, every open locally compact
subset of X is relatively pseudocompact and hence by theorem 3.2, X is ideal.

�

In the year 1990, Blair and Swardson [Proposition 2.6, [3]] proved the fol-
lowing result.

Theorem 4.10 (Proposition 2.6, [3]). A subset A of X is relatively pseudo-
compact if and only clυX A is compact.

W.W. Comfort in his paper [4, Theorem 4.1] included the following result
proved by Hager [16].

Theorem 4.11. (Hager-Johnson) Let U be open subset of X. If clυXU is
compact, then clXU is pseudocompact.

The following lemma again trivially follows from theorems 4.10 and 4.11.

Lemma 4.12. If U is an open relatively pseudocompact subset of X, then clXU
is pseudocompact.

Theorem 4.13. A space is ideal if and only if the closure of its local compact-
ness part is pseudocompact.

Proof. If X is ideal, then, the set DX of points which have compact neigh-
bourhoods is open and locally compact and hence relatively pseudocompact
by [Theorem 1.3,[2]]. So clXDX is pseudocompact. Conversely, let U be an
open locally compact subset of X. Then U ⊆ DX ⊆ clXDX. As clXDX is
pseudocompact, U is relatively pseudocompact. Hence X is ideal as follows
from theorem 3.2. �

We already mentioned in the introductory section that Henriksen and Ray-
burn in [10] did not give any example of nearly pseudocompact, which is not
anti-locally realcompact. Here we shall produce an example of nearly pseudo-
compact space which is not anti-locally realcompact.

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Ideal spaces

Example 4.14. We take any anti-locally realcompact space X. Attach (0, 0)
of the Tychonoff plank with a point y (say) in X. The resulting space is not
anti-locally realcompact as its local compact part is T \{(0, 0)} which is also
locally realcompact part; that is the space satisfies RCC property. Moreover
its almost local compact part is T which is pseudocompact and hence an ideal
space. This space is not even ∞-compact. But the above theorem 4.5 tells that
this space is nearly pseudocompact.

5. Few Properties of ideal spaces

Theorem 5.1. Regular closed subspace of an ideal space is ideal.

Definition 5.2 (A.H. Stone, [15]). A space X is called feebly compact if every
pairwise disjoint locally finite family of open sets of X is finite.

I. Glicksberg in [8] proved that in a completely regular space, the notion of
feebly compact and pseudocompact are identical. In fact he proved that pseu-
docompact completely regular space is feebly compact and a feebly compact
space is pseudocompact. In the same paper he has further shown that in a fee-
bly compact space, closure of any open set is also feebly compact. So through
chronological arguments, we conclude that within the class of completely regu-
lar spaces, regular closed subspace of a pseudocompact space is pseudocompact
and hence,in particular, in a completely regular pseudocompact space, the clo-
sure of an open set is also pseudocompact.

Proof. Let X be an ideal space. A be a regular closed subspace of X. So
clX(intXA) = A. Let DA be the set of points in A having compact neighbor-
hood in A. So DA is open subset of A. As intXA is dense in A, DA∩intXA 6= ∅.
But DA being open in A, DA ∩ intXA open in intXA and hence it is open in
X. Let x ∈ DA ∩ intXA. As x ∈ DA, x ∈ Ux ⊆ Kx,where Ux is open in A,
Kx ⊆ A is compact. Again x ∈ intXA. So there exists Wx open in X such that
x ∈ Wx ⊆ A. So x ∈ Ux ∩ Wx ⊆ Ux ⊆ Kx. Now Ux ∩ Wx is open in X as Wx is
open in X. So Kx is a compact neighbourhood of x in X. Thus x ∈ DX, the set
of all points in X having compact neighbourhood in X. So DA ∩ intXA ⊆ DX.
As intXA is dense in A and DA is open in A. So DA ⊆ clA(DA ∩ intXA). But
clA(DA ∩ intXA) = clX(DA ∩ intXA). So DA ⊆ clX(DA ∩ intXA) ⊆ clXDX.

By theorem 4.13, as X is ideal, clX(DX) is pseudocompact. We denote Ω
for clXDX. As (DA ∩ intXA) is open in X, DA ∩ intXA is open in Ω. Then
clΩ(DA∩intXA) is pseudocompact. But clΩ(DA∩intXA) = clX(DA∩intXA).
So clX(DA ∩ intXA) is pseudocompact. Again DA ⊆ clX(DA ∩ intXA) ⊆ A.
Now we track down the same argument again.

Let W = clX(DA ∩ intXA) ⊆ A. As W is pseudocompact, DA being open
in A and DA ⊆ W , DA is open in W also. So clWDA is also pseudocompact.
But clWDA = clADA. So clADA is pseudocompact. So by theorem 4.13, A is
ideal. �

Theorem 5.3. Every open C-embedded subspace of an ideal space is ideal

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B. Mitra and D. Chowdhury

Proof. Let U be a open C-embedded subset of X. Let p ∈ DU ⊆ U. So p has a
compact neighbourhood K in U. As U is open in X, K turns out to be compact
neighbourhood of p in X. So p ∈ DX ∩ U. Conversely as p ∈ DX ∩ U, there
exists a compact set K such that p ∈ intXK ⊆ K. As p ∈ U, p ∈ intXK ∩ U,
which is also open in X. Due to regularity, there exists an open set W in X such
that p ∈ W ⊆ clXW ⊆ intXK ∩ U. Now W is also open in U and clXW ⊂ K
is also compact subset of U and is therefore compact neighbourhood of p in U.
So p ∈ DU. So DU = DX ∩ U. Now as X is ideal, clXDX is pseudocompact.
As any subset of pseudocompact space is relatively pseudocompact, DU being
subset of clXDX is relatively pseudocompact subset of X. As U is C-embedded
in X, DU is relatively pseudocompact subset of U also. Again DU is open in
U. Hence clUDU is pseudocompact, by above lemma 4.12. So U is ideal by
theorem 4.13. �

Theorem 5.4. Product of ideal and compact space is ideal. Conversely if
X × Y is ideal, where Y compact, then X is ideal.

Proof. Suppose X is ideal and Y is compact. Then DX × Y = DX×Y . So
clX×Y (DX × Y ) = clXDX × Y and hence is pseudocompact as clXDX and
product of compact and pseudocompact space is pseudocompact. Second part
follows immediately from the next theorem 5.6 and from the result that the
projection on X from X × Y , where Y is compact, is a perfect map. �

The following theorem is immediate.

Theorem 5.5. Finite co-product of ideal spaces is ideal.

Proof. Let X and Y be two ideal spaces. Then clX
∐
Y DX

∐
Y = clXDX ∪

clY DY and hence is pseudocompact as clXDX and clY DY are pseudocompact.
Hence X

∐

Y is ideal. �

But the result may not be true for arbitrary co-product. For that we take
a very simple example, say N, the space of natural numbers with usual topol-
ogy. Then N is indeed countable co-product of singletons. Every singleton is
compact and hence ideal. But N is popularly known to be non-ideal space.

Theorem 5.6. Let f : X → Y be a perfect map. If X is ideal, then Y is also
ideal.

Proof. We first note that f−1DY ⊆ DX ⊆ clXDX. Hence DY ⊆ f(clXDX).
Now X is ideal, clXDX is pseudocompact. As f is closed and preserves pseu-
docompactness, f(clXDX) is also pseudocompact. clY DY being regular closed
subset of f(clXDX) is also pseudocompact. Hence by theorem 4.13, Y is
ideal. �

Corollary 5.7. If a space is not ideal, then so is its absolute.

Proof. The corollary directly follows from the above theorem 5.6 as there always
exist a perfect irreducible map from the absolute of a space onto the space
itself. �

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 88



Ideal spaces

The next theorem 5.9 follows trivially from the following theorem 5.8 by
Henriksen and Rayburn [10, Theorem 3.17].

Theorem 5.8. If X and Y are nearly pseudocompact, then X × Y is nearly
pseudocompact if and only if clXDX × clY DY is pseudocompact.

Theorem 5.9. If X and Y are nearly pseudocompact spaces, then X × Y is
nearly pseudocompact if and only if X × Y is ideal.

Proof. As X × Y is nearly pseudocompact, X × Y is also ideal. Conversely,
if X × Y is ideal, then clX×Y DX×Y , is pseudocompact. But clX×Y DX×Y =
clXDX ×clY DY . So clXDX ×clY DY is pseudocompact. Hence X ×Y is nearly
pseudocompact. �

Theorem 5.10. A realcompact, ideal space is ∞-compact.

Proof. Since X is ideal, clXDX is pseudocompact. Since X is realcompact,
clXDX is also realcompact. Hence it is compact. So clearly every open locally
compact subset is bounded by a compact set. Hence X is ∞-compact. �

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© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 89