@ Appl. Gen. Topol. 16, no. 2(2015), 209-215doi:10.4995/agt.2015.3449
c© AGT, UPV, 2015

Finite products of limits of direct systems

induced by maps

Ivan Ivanšić
a
and Leonard R. Rubin

b

a
Department of Mathematics, University of Zagreb, Unska 3, P.O. Box 148, 10001 Zagreb,

Croatia (ivan.ivansic@fer.hr)
b

Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019, USA

(lrubin@ou.edu)

Abstract

Let Z, H be spaces. In previous work, we introduced the direct (inclu-
sion) system induced by the set of maps between the spaces Z and H.
Its direct limit is a subset of Z × H, but its topology is different from
the relative topology. We found that many of the spaces constructed
from this method are pseudo-compact and Tychonoff. We are going to
show herein that these spaces are typically not sequentially compact
and we will explore conditions under which a finite product of them
will be pseudo-compact.

2010 MSC: 54B35; 54C55; 54C20; 54D35; 54F45.

Keywords: complete regularity; convergent sequence; direct limit; di-
rect system; inclusion direct system; normality; perfect space;
pseudo-compact; regularity; sequential convergence; sequential
extensor.

1. Introduction

In [2] we introduced the concept of the direct system X induced by the
set of maps F = C(Z, H) between spaces Z and H, and in [3] we extended
these ideas to include systems induced by a perhaps proper nonempty subset
F of C(Z, H). This system X consists of all the unions of the graphs of finite
nonempty subsets of F and the inclusion maps induced when one finite subset
is contained in another. Such an entity is an “inclusion” direct system. Its
direct limit Xδ = dirlim X, as a set, is the union of the graphs of the elements

Received 11 December 2014 – Accepted 20 March 2015

http://dx.doi.org/10.4995/agt.2015.3449


I. Ivanšić and L. R. Rubin

of F and hence Xδ ⊂ Z × H as a set. But the topology of Xδ might be larger
than that induced by Z × H. In [2] we were concerned mainly (Theorems
5.1, 6.1(2)) with determining when Xδ is pseudo-compact. By a space being
pseudo-compact, we mean that every map of it to R has bounded image (see
[1] where additional, unnecessary requirements are placed on such a space). In
[3] we considered pseudo-compactness and other properties such as regularity,
complete regularity, and normality. There we proved the existence of a large,
but abstract class of pseudo-compact spaces. We are going to show herein that
these spaces are typically not sequentially compact but that finite products of
them are pseudo-compact. Our main result is Theorem 2.13 which shows that
finite products of these spaces are pseudo-compact. Whether infinite products
are pseudo-compact is an open question.

2. Products of Pseudo-compact Spaces

In Section 4 of [4], there is a discussion about products of pseudo-compact
spaces. For example it is pointed out that even the product of two pseudo-
compact spaces need not be pseudo-compact. A useful fact from this reference
is its Theorem 4.1.

Theorem 2.1. Let {Xa | a ∈ A} be a collection of spaces. If X =
∏
{Xa | a ∈

A} is pseudo-compact, then for every subset B of A,
∏
{Xa | a ∈ B} is pseudo-

compact. If A is infinite and X is not pseudo-compact, then there is a countably

infinite subset J of A such that
∏
{Xa | a ∈ J} is not pseudo-compact.

Theorem 4.4 of [4] asserts the following.

Proposition 2.2. Every product of pseudo-compact spaces of which all but one

are sequentially compact is pseudo-compact.

For example, let Ω denote the first uncountable ordinal. The first uncount-
able ordinal space [0, Ω) is not compact, but it is pseudo-compact and sequen-
tially compact, so any product of copies of this space is pseudo-compact. Yet,
sequential compactness is not a necessary condition for such an outcome. Let
ω be the first infinite ordinal. Then we have the next result.

Example 2.3. Let X denote the space [0, Ω] × [0, ω] \ {(Ω, ω)}. Then X is
not sequentially compact, but Xω is pseudo-compact and hence X is pseudo-
compact.

Proof. Clearly X is not sequentially compact. It is not difficult to see that
for each nonempty open subset U of Xω, there exists w = (w1, w2, . . . ) ∈ U
such that for each i the first coordinate of wi does not equal Ω. If X

ω is not
pseudo-compact, then we may choose a map f : Xω → R and a set {xi | i ∈ N}
in Xω such that f(xi) = i for each i. Let Vi = (i −

1
3
, i + 1

3
) and Wi = f

−1(Vi).
For each i, choose yi = (yi,1, yi,2, . . . ) ∈ Wi so that for each j, the first

coordinate zi,j of yi,j does not equal Ω. The countable collection of these zi,j
has an upper bound α < Ω. It follows that {yi | i ∈ N} ⊂ ([0, α] × [0, ω])

ω, a
compact subset of Xω. This shows that the unbounded set {f(yi) | i ∈ N} is
contained in a compact subset of R, a contradiction. �

c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 2 210



Finite products of limits of direct systems induced by maps

In [3] we proved the existence of a large class of pseudo-compact spaces,
which are the direct limits of certain direct systems. Let us review the main
ideas surrounding the construction of such direct systems. For the remainder of
this section, let Z and H be nonempty Hausdorff spaces, and C(Z, H) denote
the set of maps of Z to H. Fix a nonempty subset F of C(Z, H) and let A be
the collection of nonempty finite subsets of F ordered by inclusion, which we
denote �. Thus, (A, �) is a directed set. Recall that whenever f ∈ C(Z, H),
then Gf ⊂ Z × H, the graph of f, is closed in Z × H.

Definition 2.4. For each a ∈ A, let Xa =
⋃
{Gf | f ∈ a} ⊂ Z × H. Whenever

a � b, let pba : Xa → Xb denote the inclusion map.

Lemma 2.5. The system X = (Xa, p
b
a, (A, �)) is a direct system of closed

inclusion maps and Hausdorff spaces Xa. Let Xδ = dirlim X. Then, as a set,
Xδ =

⋃
{Xa | a ∈ A} ⊂ Z × H, but Xδ has the weak topology determined by

{Xa | a ∈ A}. The set inclusion ιXδ : Xδ →֒ Z × H is a map in conjunction
with the respective topologies of Xδ and the product topology of Z × H. �

Definition 2.6. Let F be a nonempty subset of C(Z, H), A be the collection
of nonempty finite subsets of F, and X = (Xa, p

b
a, (A, �)) be as in Lemma 2.5.

Then we call X the inclusion direct system induced by F.

Such systems as in Definition 2.6 frequently have direct limits Xδ that are
pseudo-compact. We state Theorem 4.1 of [3].

Theorem 2.7. Let Z, H be nonempty spaces, X = (Xa, p
b
a, (A, �)) be the

inclusion direct system induced by a nonempty subset F of C(Z, H), and Xδ =
dirlim X. Suppose that:

(1) both Z and H are sequentially compact,
(2) H is a sequential extensor modulo F for Z,
(3) Z and H are Hausdorff spaces,
(4) Z is a perfect space,
(5) Z is a first countable space, and
(6) either F is dense for C(Z, H), or H is first countable.

Then Xδ is pseudo-compact.

There are two notions in this theorem which should be defined. Before those,
we also need the next idea.

Definition 2.8. Let M = {1
i
| i ∈ N} ∪ {0} ⊂ R. We shall refer to M as the

convergent sequence. Any space homeomorphic to the convergent sequence
will be called a convergent sequence.

Definition 2.9. Let F ⊂ C(Z, H). We will say that H is a sequential
extensor modulo F for Z if for each convergent sequence M = {zi | i ∈
N}∪{z} in Z, sequence (fi) in F such that the sequence (fi(zi)) converges to an
element w ∈ H, and sequence (Ui) of neighborhoods Ui of (zi, fi(zi)) in Z ×H,
there exist f ∈ F and a subsequence (zij ) of (zi) such that (zij , f(zij )) ∈ Uij
for all j.

c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 2 211



I. Ivanšić and L. R. Rubin

Definition 2.10. Let X = (Xa, p
b
a, (A, �)) be the inclusion direct system

induced by a nonempty subset F of C(Z, H) and Xδ = dirlim X. Suppose that
for each sequence (zi)i∈N in Z converging to an element z ∈ Z, h ∈ C(Z, H)
with (z, h(z)) ∈ Xδ, and neighborhood U of (z, h(z)) in Xδ, there exist g ∈
C(Z, H) and i ∈ N such that,

(1) g ∈ F,
(2) (zi, g(zi)) ∈ U, and
(3) g(zi) = h(z).

Then we shall say that F is dense for C(Z, H).

To see how easily such conditions as in Theorem 2.7 can be made to occur,
let Z = [0, 1] = H. Then Example 7.3 of [3] along with an application of
Lemma 7.2 of [3] yield the following.

Example 2.11. Let F = C(Z, H), X = (Xa, p
b
a, (A, �)) be the inclusion

direct system induced by F, and Xδ = dirlim X. Then Xδ is a completely
regular, pseudo-compact Hausdorff space, but Xδ is neither compact, normal,
nor sequentially compact.

We are going to show in Theorem 2.13 that finite products of spaces Xδ such
as those in Theorem 2.7 are pseudo-compact. We need a lemma in support of
the proof of that theorem.

Lemma 2.12. Suppose that X = (Xa, p
b
a, (A, �)) satisfies the hypothesis of

Theorem 2.7 and Xδ = dirlim X. Then for each sequence (Ui) of nonempty
open subsets of Xδ, there exist:

(1) f ∈ F,
(2) z ∈ Z,
(3) a subsequence (Uij ) of (Ui), and
(4) for all j ∈ N, a point zj ∈ Z

such that for each j, (zj, f(zj)) ∈ Uij , (zj) converges to z ∈ Z, and (f(zj))
converges to f(z) in H. Hence, (zj, f(zj)) converges to (z, f(z)) in Gf ⊂ Xδ.

Proof. Let (Ui) be a sequence of nonempty open subsets of Xδ. For each i ∈ N,
choose ai ∈ A, fi ∈ ai, and zi ∈ Z such that xi = (zi, fi(zi)) ∈ Ui. Using (1)
of 2.7 and passing to a subsequence, if necessary, we may as well assume that
the sequence (fi(zi))i∈N in H converges to some element w ∈ H. Applying (1)
of 2.7 again, we may also assume that the sequence (zi)i∈N in Z converges to
an element z ∈ Z.

There are two cases to consider, either {zi | i ∈ N} is a finite set or it is
an infinite set. Let us show that if it is finite, then we can replace it with
an infinite set satisfying the above conditions, independently of which of the
two parts of 2.7(6) is in play. Using (3) of 2.7 and passing to a subsequence if
necessary, we may assume that for each i, zi = z. Making use of (3)–(5) of 2.7,
choose a sequence (z∗i )i∈N in Z converging to z so that for i 6= j, z

∗

i 6= z
∗

j , and
for each i, z∗i 6= z.

c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 2 212



Finite products of limits of direct systems induced by maps

Our plan is to show that (6) of 2.7 allows us to find a sequence i1 < i2 < · · ·
in N along with a sequence (gj) so that for each j, bj = {gj} ∈ A, x

∗

j =
(z∗ij , gj(z

∗

ij
)) ∈ Uj, and the sequence (gj(z

∗

ij
)) converges in H to w. If that can

be done, we will define Q∗ = {x∗j | j ∈ N}. Then for each j ∈ N: bj ∈ A, gj ∈ bj,
z∗ij ∈ Z, and x

∗

j = (z
∗

ij
, gj(z

∗

ij
)) ∈ Uj. The sequence (z

∗

ij
)j∈N converges to z. The

reader will now observe that we may replace (zj, xj, aj, fj) by (z
∗

ij
, x∗j , bj, gj),

j = 1, 2, . . . , to satisfy the conditions of the first paragraph of this proof. But
this time {z∗ij | j ∈ N} is an infinite set.

We now show that whichever part of 2.7(6) is in operation, we can produce
such a collection of elements (z∗ij , x

∗

j , bj, gj), j = 1, 2, . . . , as just described. If,

(Case 1), F is dense for C(Z, H), then define Vi = Ui for all i. If, (Case 2),
F is not dense for C(Z, H), and hence H is first countable, we define the sets
Vi differently. Let {Qi | i ∈ N} be a local base for the neighborhood system
of w in H with Qi+1 ⊂ Qi for all i. Since (fi(z)) converges to w, we may
assume that fi(z) ∈ Qi for each i. Define Vi = (Z × Qi) ∩ Ui, noting that
this equals (Z × Qi) ∩ Xδ ∩ Ui. Since ιXδ : Xδ → Z × H is a map, then
ι
−1
Xδ

(Z × Qi) ∩ Ui = (Z × Qi) ∩ Xδ ∩ Ui is open in Xδ for each i. Thus Vi is

an open neighborhood of xi = (zi, fi(zi)) = (z, fi(z)) in Xδ in either case of
2.7(6).

Under the assumption of (Case 1), apply Definition 2.10 with (zi)i∈N re-
placed by (z∗i )i∈N, h = f1, (z, h(z)) = (z, f1(z)), and U = V1. Using (1)–(3) of
that definition, we obtain i1 ∈ N and g1 ∈ F such that x

∗

1 = (z
∗

i1
, g1(z

∗

i1
)) ∈

V1 = U1 and g1(z
∗

i1
) = f1(z). If (Case 2) prevails, put g1 = f1 ∈ F; we shall

show, as in (Case 1), that there exists i1 ∈ N so that x
∗

1 = (z
∗

i1
, g1(z

∗

i1
)) ∈ V1.

Since (z∗i ) converges to z ∈ Z, then (g1(z
∗

i )) converges to g1(z) in H. Note that
Gg1 is a closed subspace of both Z × H and Xδ. So the sequence (z

∗

i , g1(z
∗

i ))
converges in Gg1 to (z, g1(z)). Since V1 ∩ Gg1 is a neighborhood of (z, g1(z)) =
(z, f1(z)) in Gg1, then for some i1, (z

∗

i1
, g1(z

∗

i1
)) ∈ V1. So in either case, there

exist g1 and i1 so that {g1} ∈ A and (z
∗

i1
, g1(z

∗

i1
)) ∈ V1, but in the first case we

also have that (z∗i1, g1(z
∗

i1
)) = (z∗i1, f1(z)).

Replace the sequence (z∗i )i∈N by (z
∗

i )i>i1 . Using the latter, apply the same
technique we just employed, this time with (f2, x2) replacing (f1, x1), to find
i2 > i1 and g2 ∈ F such that x

∗

2 = (z
∗

i2
, g2(z

∗

i2
)) ∈ V2 ⊂ U2 with the addi-

tional property that g2(z
∗

i2
) = f2(z) in (Case 1). Such a process is to be done

recursively so that we find a sequence i1 < i2 < · · · in N, a sequence (gi)i∈N
in F, bi = {gi} ∈ A, and a subsequence (z

∗

ij
)j∈N of (z

∗

i )i∈N such that for each

j ∈ N, x∗j = (z
∗

ij
, gj(z

∗

ij
)) ∈ Vj ⊂ Uj, and, in (Case 1), gj(z

∗

ij
) = fj(z). In

the latter case, the sequence (gj(z
∗

ij
))j∈N equals (fj(z))j∈N = (fj(zj))j∈N, so it

converges to w ∈ H. In (Case 2), x∗j = (z
∗

ij
, gj(z

∗

ij
)) ∈ Vj ⊂ Z ×Qj. This shows

that gj(z
∗

ij
) ∈ Qj, and hence the sequence (gj(z

∗

ij
))j∈N converges to w ∈ H.

We have proved that under the assumption 2.7(6), a sequence (z∗ij , x
∗

j , bj, gj),

j = 1, 2, . . . , can be produced with {z∗ij | j ∈ N} an infinite set as requested

above.

c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 2 213



I. Ivanšić and L. R. Rubin

We assume therefore that {zi | i ∈ N} is an infinite set. By passing to a
subsequence of (zi)i∈N if necessary we may stipulate that for i 6= j, zi 6= zj and
that for each i, zi 6= z. By 2.7(3), Z is Hausdorff, so B = {zi | i ∈ N} ∪ {z}
is a convergent sequence in the sense of Definition 2.8. Define λ : B → H by
λ(zi) = fi(zi), and λ(z) = w. Then λ is a map. Applying 2.7(2) get a map
f : Z → H such that c = {f} ∈ A along with a subsequence (zij ) of (zi) such
that (zij , f(zij )) ∈ Uij for all j. Since A is a collection of subsets of F, then
f ∈ F. �

Now we can prove our main result about the preservation pseudo-compactness
in any finite product of the above types of spaces.

Theorem 2.13. Let n ∈ N and Xδ1, . . . , Xδn be spaces like those obtained in
Theorem 2.7. Then X = Xδ1 × · · · × Xδn is pseudo-compact.

Proof. For simplicity we will prove this in case n = 2; the reader will easily
fill in the details needed for the general case. Let (Fk, Zk, Hk) correspond to
(F, Z, H) in Lemma 2.12 for k ∈ {1, 2}.

Suppose that X is not pseudo-compact. Then there exist a subset {xi | i ∈ N}
of X and a map ω : X → R such that ω(xi) = i for each i. Put Wi = (i−

1
3
, i+ 1

3
)

and let Qi = ω
−1(Wi). For each i, find open sets U

k
i of Xδk , k ∈ {1, 2}, such

that xi ∈ U
1
i × U

2
i ⊂ Qi.

First apply Lemma 2.12 to the nonempty open subsets U1i of Xδ1. There
exist f1 ∈ F1, z1 ∈ Z1, a subsequence (U

1
ij
) of (U1i ), and for all j ∈ N, a point

z1j ∈ Z1 such that for each j, (z
1
j , f1(z

1
j )) ∈ U

1
ij
, (z1j ) converges to z1 ∈ Z1, and

(f1(z
1
j )) converges to f1(z1) in H1. Hence, (z

1
j , f1(z

1
j )) converges to (z1, f1(z1))

in Gf1 ⊂ Xδ1.
By passing to a subsequence we may as well assume that U1ij = U

1
j for each

j. Now apply the same procedure to the nonempty open sets U2i of Xδ2. We
get f2 ∈ F2, z2 ∈ Z2, a subsequence (U

2
ij
) of (U2i ), and for all j ∈ N, a point

z2j ∈ Z2 such that for each j, (z
2
j , f2(z

2
j )) ∈ U

2
ij
, (z2j ) converges to z2 ∈ Z2, and

(f2(z
2
j )) converges to f2(z2) in H2. Hence, (z

2
j , f2(z

2
j )) converges to (z2, f2(z2))

in Gf2 ⊂ Xδ2.
By passing to subsequences we may assume the following. There are se-

quences (zki ) in Zk, k ∈ {1, 2}, such that (fk(z
k
i )) converges to fk(zk) in Gfk

with (r1i , r
2
i ) ∈ U

1
i × U

2
i ⊂ Qi for each i where we define r

k
i = (z

k
i , fk(z

k
i )). Let

Dk = {r
k
i | i ∈ N} ∪ {(zk, fk(zk))}. Then each Dk ⊂ Gfk ⊂ Xδk is compact.

Now, ω(r1i , r
2
i ) ∈ ω(Qi) ⊂ Wi for each i. So ω({(r

1
i , r

2
i ) | i ∈ N}) ⊂ ω(D1 × D2),

a compact subset of R, which is impossible. �

c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 2 214



Finite products of limits of direct systems induced by maps

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360–367.
[3] I. Ivanšić and L. Rubin, The topology of limits of direct systems induced by maps,

Mediterr. J. Math. 11, no. 4 (2014), 1261–1273.
[4] R. M. Stephenson, Jr., Pseudocompact spaces, Trans. Amer. Math. Soc. 134 (1968),

437–448.

c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 2 215