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@ Appl. Gen. Topol. 17, no. 1(2016), 1-5doi:10.4995/agt.2016.4616
c© AGT, UPV, 2016

The equivalence of two definitions of

sequential pseudocompactness

Paolo Lipparini

a
Dipartimento di Matematica, Viale della Ricerca Scient̀ıfica, II Università Gelmina di Roma

(Tor Vergata), I-00133 Rome, Italy (lipparin@axp.mat.uniroma2.it)

Abstract

We show that two possible definitions of sequential pseudocompactness

are equivalent, and point out some consequences.

2010 MSC: Primary 54D20; Secondary 54B10.

Keywords: sequentially pseudocompact; feebly compact; topological space.

1. The equivalence

According to Artico, Marconi, Pelant, Rotter and Tkachenko [1, Definition
1.8], a Tychonoff topological space X is sequentially pseudocompact if the fo-
llowing condition holds.

(1) For any family (On)n∈ω of pairwise disjoint nonempty open sets of
X, there are an infinite set J ⊆ ω and a point x ∈ X such that every
neighborhood of x intersects all but finitely many elements of (On)n∈J.

Notice that in [1] X is assumed to be a Tychonoff space, but the above
definition makes sense for an arbitrary topological space.

According to Dow, Porter, Stephenson, and Woods [2, Definition 1.4], a
topological space is sequentially feebly compact if the following condition holds.

(2) For any sequence (On)n∈ω of nonempty open subsets of X, there are
an infinite set J ⊆ ω and a point x ∈ X such that every neighborhood
of x intersects all but finitely many elements of (On)n∈J.

(the difference is that in Condition (1) the On’s are assumed to be pairwise
disjoint, while they are arbitrary in Condition (2))

Received 10 October 2013 – Accepted 11 September 2015

http://dx.doi.org/10.4995/agt.2016.4616


P. Lipparini

The above two notions have been rather thoroughly studied by the men-
tioned authors. In this note we show their equivalence. Putting together the
results from [1] and [2] shows that the class of sequentially pseudocompact
Tychonoff topological spaces is closed under (possibly infinite) products and
contains significant classes of pseudocompact spaces.

Unless otherwise specified, we shall assume no separation axiom.

Theorem 1.1. For every topological space X, Conditions (1) and (2) above
are equivalent.

Proof. Condition (2) trivially implies Condition (1).
For the converse, suppose that X satisfies Condition (1), and let (On)n∈ω

be a sequence of nonempty open sets of X. Suppose by contradiction that

(*) for every infinite set J ⊆ ω and every point x ∈ X there is some
neighborhood U(J,x) of x such that N(J,x) = {n ∈ J | U(J,x)∩On =
∅} is infinite.

Without loss of generality, we can assume that U(J,x) is open. We shall
construct by simultaneous induction a sequence (mi)i∈ω of distinct natural
numbers, a sequence (Ji)i∈ω of infinite subsets of ω, and a sequence of pairwise
disjoint nonempty open sets (Ui)i∈ω such that

(a) Ui ⊆ Omi for every i ∈ ω,
(b) Ui ∩ On = ∅, for every i ∈ ω, and n ∈ Ji, and
(c) Ji ⊇ Jh, whenever i ≤ h ∈ ω.

Put m0 = 0 and pick x0 ∈ O0 (this is possible, since O0 in nonempty). Apply
(*) with J = ω and x = x0, and let U0 = U(ω,x0) ∩ O0 ⊆ O0 = Om0 and
J0 = N(ω,x0). U0 is nonempty, since x0 ∈ U(ω,x0)∩O0. By (*), J0 is infinite,
and Clause (b) is satisfied for i = 0. The basis of the induction is completed.

Suppose now that 0 6= i ∈ ω, and that we have constructed finite sequences
(mk)k<i, (Jk)k<i, and (Uk)k<i satisfying the desired properties. Let mi be any
element of Ji−1. Since Ji−1 is infinite, we can choose mi distinct from all the
mk’s, for k < i (however, this follows automatically from (a) - (c)). Let xi
be any element of the nonempty Omi. Apply (*) with J = Ji−1 and x = xi,
and let Ui = U(Ji−1,xi) ∩ Omi ⊆ Omi and Ji = N(Ji−1,xi). As above, Ui
is nonempty, since xi ∈ U(Ji−1,xi) ∩ Omi. By the definition of N(Ji−1,xi),
we have that Ji ⊆ Ji−1, hence Clause (c) holds, by the inductive hypothesis.
By (*), Ji is infinite, and moreover Clause (b) is satisfied for i. It remains to
show that Ui is disjoint from Uk, for k < i. Since, by construction, mi ∈ Ji−1,
then, by (c) of the inductive hypothesis, for every k < i, we have that mi ∈ Jk,
hence, by (b), Uk ∩ Omi = ∅, hence also Uk ∩ Ui = ∅, since by construction
Ui ⊆ Omi. The induction step is thus complete.

Having constructed sequences satisfying the above properties, we can apply
Condition (1) to the sequence (Ui)i∈ω of nonempty pairwise disjoint open sets,
getting some J ⊆ ω and some x ∈ X such that every neighborhood of x
intersects all but finitely many elements of (Ui)i∈J. If we put J

∗ = {mi | i ∈
J}, then every neighborhood of x intersects all but finitely many elements of

c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 1 2



The equivalence of two definitions of sequential pseudocompactness

(On)n∈J∗ , because of Clause (a). We have reached a contradiction, thus the
theorem is proved. �

2. Consequences

Putting together results from [1, 2], and using Theorem 1.1, we get some
nice results about sequential pseudocompactness. First, we recall the following
results from [2].

Theorem 2.1.

[2, Theorems 4.1 and 4.4] A product of nonempty topological spaces is sequen-
tially feebly compact if and only if each factor is sequentially feebly compact.

[2, Theorem 4.3] Every product of feebly compact spaces, all but one of which
are sequentially feebly compact, is feebly compact.

Recall that a topological space is feebly compact if, for any sequence (On)n∈ω
of nonempty open sets of X, there is a point x ∈ X such that {n ∈ ω | U ∩On 6=
∅} is infinite, for every neighborhood U of x. Clearly, Condition (2) implies
feeble compactness. It is well known that a Tychonoff space is feebly compact
if and only if it is pseudocompact.

From Theorem 2.1 we immediately get:

Corollary 2.2. Let P be a family of properties of topological spaces such that
every feebly compact topological space satisfying at least one P ∈ P is sequen-
tially feebly compact. Then every product of feebly compact spaces, all but one
of which satisfy some P ∈ P, is feebly compact (pseudocompact, if the product
is Tychonoff).

Corollary 2.2 is interesting since, restricted to Tychonoff spaces, and using
Condition (1), [1] found many properties P which satisfy the assumption in
Corollary 2.2; among them, the properties of being a topological group, or a
scattered space, or a first countable space, or a ψ-ω-scattered space. From The-
orem 1.1 and Corollary 2.2 we get a proof that any product of pseudocompact
Tychonoff spaces satisfying one of the above properties is pseudocompact. In
particular, from Theorems 1.1, 2.1 and [1, Proposition 1.10], we get another
proof of the classical result by Comfort and Ross that any product of Tychonoff
pseudocompact topological groups is pseudocompact. However, it is not clear
whether this is a real simplification: it might be the case that any proof that
every Tychonoff pseudocompact topological group is sequentially pseudocom-
pact already contains enough sophistication to be easily converted into a direct
proof of Comfort and Ross Theorem.

The above considerations lead to the following problem.

Problem 2.3. Is there some significant part of the (topological) theory of pseu-
docompact topological groups which follows already from the assumption of se-
quential pseudocompactness?

More precisely, are there other theorems holding for pseudocompact topolog-
ical groups which can be generalized to sequentially pseudocompact topological
spaces (with not necessarily some algebraic structure on them)?

c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 1 3



P. Lipparini

3. Further remarks

The relationship between sequential feeble compactness and sequential com-
pactness does not always parallel the relationship between feeble compactness
and countable compactness. Both βω and Dc are classical examples of compact
not sequentially compact spaces. As noticed on [1, p. 7] and [2, Example 2.9],
βω is not sequentially pseudocompact. On the other hand, Dc is sequentially
pseudocompact, by Theorem 2.1. Thus, compactness together with sequen-
tial pseudocompactness do not necessarily imply sequential compactness. In
particular, normality and sequential pseudocompactness do not imply sequen-
tial compactness (thus the result that normality and pseudocompactness imply
countable compactness cannot be generalized in the obvious way). Also, a com-
pact subspace of a compact sequentially pseudocompact space is not necessarily
sequentially pseudocompact, since βω can be embedded in Dc. In particular,
a closed subspace of a sequentially pseudocompact space is not necessarily se-
quentially pseudocompact.

The proof of [2, Theorem 4.1] actually shows a little more. If (Xh)h∈H is a
family of topological spaces, the ω-box topology on

∏
h∈H

Xh is defined as the
topology a base of which consists of the sets of the form

∏
h∈H

Oh, where each
Oh is an open set of Xh, and |{h ∈ H | Oh 6= Xh}| ≤ ω.

Proposition 3.1. Suppose that (Xh)h∈H is a family of sequentially feebly com-
pact topological spaces. If (On)n∈ω is a sequence of nonempty open sets in the
ω-box topology on

∏
h∈H

Xh, then there are an infinite set J ⊆ ω and a point
x ∈

∏
h∈H

Xh such that {n ∈ J | U ∩On = ∅} is finite, for every neighborhood
U of x in the Tychonoff product topology on

∏
h∈H

Xh.

Proof. Same as the proof of [2, Theorem 4.1], since it is no loss of generality
to deal with elements of the standard base of the ω-box topology, and since
the set R defined as in the proof of [2, Theorem 4.1] is countable in this case,
too, being the countable union of a family of countable sets. See [3] for full
details. �

In the statement of Proposition 3.1, the neighborhoods U of x have to be
considered in the Tychonoff product topology. The statement would turn out to
be false allowing U vary among the neighborhoods of x in the ω-box topology.
Indeed, for example, a discrete two-element space is vacuously sequentially
pseudocompact; however, its ωth power in the ω-box topology is a discrete
space of cardinality c, hence the conclusion of Proposition 3.1 would fail.

The following notions might deserve some study, in particular when α is a
cardinal.

Definition 3.2. For α an infinite limit ordinal, we say that a topological space
X is sequentially α-feebly compact if, for any sequence (Oβ)β∈α of nonempty
open sets of X, there are some x ∈ X and a subset Z of α such that Z has
order type α, and, for every neighborhood U of x, there is β < α such that
U ∩ Oγ 6= ∅, for every γ ∈ Z such that γ > β.

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The equivalence of two definitions of sequential pseudocompactness

If we modify the above definition by further requesting that the (Oβ)’s are
pairwise disjoint, we say that X is d-sequentially α-feebly compact. Clearly, for
every α, sequential α-feeble compactness implies d-sequential α-feeble compact-
ness, and, for α = ω, both notions are equivalent (and equivalent to sequential
feeble compactness), by Theorem 1.1.

Acknowledgements. We wish to express our gratitude to X. Caicedo and S.

Garćıa-Ferreira for stimulating discussions and correspondence. We thank our

students from Tor Vergata University for stimulating questions.

References

[1] G. Artico, U. Marconi, J. Pelant, L. Rotter and M. Tkachenko, Selections and suborder-
ability, Fund. Math. 175 (2002), 1-33.

[2] A. Dow, J. R. Porter, R. M. Stephenson, Jr. and R. G. Woods, Spaces whose pseudo-
compact subspaces are closed subsets, Appl. Gen. Topol. 5 (2004), 243-264.

[3] P. Lipparini, Products of sequentially pseudocompact spaces, arXiv:1201.4832 (2012).

c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 1 5