The Second Local Conference of Mathematical Science and Applications,

Behbahan Khatam Alanbia University of Technology, 5 Dey, 1397 (December 26, 2018)

ON the product of λ -compact spaces

M. Namdari
Shahid Chamran university of Ahvaz, namdari@ipm.ir

M. A. Siavoshi
Shahid Chamran university of Ahvaz, m.siavoshi@scu.ac.ir

Abstract
In this article some properties of λ -compact spaces are mentioned and it is shown that the
product of two λ -compact Pλ -spaces is a λ -compact space.

Keywords: λ -compact, Pλ -space, product of Pλ -spaces.

MSC(2010): Primary: 54A25, 54C30; Secondary: 13J30, 13J25.

1 Introducton
λ -compact spaces with various definitions are stated in the literature, see [3], [7] and [8]. In this
article, we first give our definition of λ -compact spaces, which is given in [10], and then try to
extend some of the important results which hold in compact, Lindelöf spaces to general λ -compact
spaces.
Throughout this paper all spaces are assumed to be T2 and λ is an infinite regular cardinal num-
ber. The reader is referred to [2], [5] for possible undefined terms and notations.
Tychonoff theorem states that the product of any family of compact spaces is compact. We know
that this theorem is One of the most important results in topology. It is natural to try to generalize
this result to Lindelöf spaces, and λ -compact spaces. Unfortunately the Sorgenfrey Line gives us a
counterexample regarding this. In fact, the product of to Lindelöf spaces is not Lindelöf in general.
In [] Misra has shown that the product of two lindelöf p-spaces is lindelöf. In this paper we show
that product of two λ -compact pλ -spaces is λ -compac.

1
2 λ -compact spaces
Definition 2.1. A topological space X is said to be λ -compact whenever each open cover of X can
be reduced to an open cover of X whose cardinality is less than λ , where λ is the least infinite
cardinal number with this property. In this case we say that λ is the compactness degree of X and
write dc (X) = λ .
If |X| = γ and for each regular cardinal number β < γ the space X is not β -compact then it
is either γ-compact or γ + -compact, where, γ + is the least regular cardinal number greater than γ.
Hence every topological space X is µ-compact for some regular cardinal µ. Clearly if |X| < λ then
dc (X) ≤ λ . Therefore in the study of the λ -compact spaces we consider |X| ≥ λ .
We note that ℵ◦ -compact spaces, ℵ1 -compact spaces are exactly compact and non-compact
lindelöf spaces respectively. The following example which is barrowed from [8], shows that for every
regular cardinal number λ , there exists a λ -compact Hausdorff space, which is in fact a completely
regular space too.
Example 2.2. Let (X, T ) be a discrete topological space with |X| > λ where λ is a regular cardinal
number. Suppose that a ∈
/ X and take:

X ∗ = X ∪ {a} & T ∗ = T ∪ {Ga ⊆ X ∗ : a ∈ Ga , |X ∗ \ Ga | < λ }.

We show that (X ∗ , T ∗ ) is a λ -compact Hausdorff space.

Let {Gi }i∈I be a open cover of X ∗ such that |I| > λ . There exists k ∈ I such that a ∈ Gk ; hence
|X ∗ \ Gk | < λS. Since (X ∗ \ Gk ) ⊆ j6=k G j , there exists J ⊆ I with cardinality less than λ such that
S

(X ∗ \ Gk ) ⊆ j∈J G j .
X ∗ = (X ∗ \ Gk ) Gk = ( G j ) Gk .
[ [ [

j∈J

Hence we have already shown that X∗ is β -compact space for some β ≤ λ . We claim that β < λ
leads us to a contradiction. Now let β < λ and consider a subset F ( X with |F| = β and take
Ga = X ∗ \ F. Clearly
|X ∗ \ Ga | = |F| = β < λ ⇒ Ga ∈ (T ∗ \ T ).
But {Ga } {{x}}x∈F is an open cover of X ∗ with cardinality β which has no subcover with cardi-
S

nality less than β and this is the desired contradiction. Finally, it remains to be shown that X ∗ is
Hausdorff. Since X is Hausdorff and T ⊆ T ∗ , every two distinct elements of X have disjoint neigh-
borhoods in X and consequently in X ∗ . Now suppose x 6= a is a member of X ∗ , then X ∗ \ {x} and
{x} are two disjoint open sets which separate x and a in X ∗ . Clearly, (X ∗ , T ∗ ) has a base consisting
of clopen sets, hence it is completely regular.
Proposition 2.3. Let B be a base for the topological space X. the space X is λ -compact if and
only if every cover of X, consisting of some of the elements of B, can be reduced to a subcover with
cardinality less than λ .
S
Proof. The sufficiency is obvious. For necessary, let X = k∈K Gk , where Gk is an open subset of X
for every k ∈ K. Suppose B = {Bi }i∈I . Since B is a base for X, we have:
[ [
X= ( Bik ).
k∈K ik ∈Jk ⊆I

2
Consequently {Bik : k ∈ K, ik ∈ I} ⊆ B is an open cover of X. So by our hypothesis it has the
subcover {Bikr }r∈R such that its cardinality is less than λ . Clearly
[ [
X= Bikr = Gkr
r∈R r∈R

and clearly the cardinality of {Gkr }r∈R is less than λ .

Definition 2.4. We say that the topological space X is λ -normal, whenever, for every closed subset
F ⊆ X and every open subset W ⊆ X containing F there exists a family {Wα }α∈Λ of open sets with
|Λ| < λ such that [
∀α ∈ Λ, W α ⊆ W , F ⊆ Wα .
α∈Λ

Note that every ℵ1 -normal space is a normal space, see [2, 1.5.15].

The following general fact is interesting.

Proposition 2.5. If F is a closed subspace of the λ -compact space X then dc (F) ≤ λ .

Proof. Let F ⊆ i∈I Gi , where Gi is an open set for each i ∈ I. Hence G ) (X \ F).
S S S
X= (
S i∈I i
SinceSX is λ -compact, there exists J ⊆ I whit |J| < λ such that X = ( i∈J Gi ) (X \ F). Obviously,
S

F ⊆ i∈J Gi . This means that dc (F) ≤ λ .

Definition 2.6. we say that the topological space X is a Pλ -space, whenever every Gλ -set in it is open
(note, every arbitrary topological space is a Pℵ◦ -space and X is a P-space if and only it is a Pℵ1 -space.

The next result is the generalization of the well-known fact that every compact space is regular.
Proposition 2.7. Every λ -compact Pλ -space is regular.
Proof. Let F be a closed subset of the λ -compact Pλ -space X and x ∈ X \F. For any y ∈ F there exist
disjoint open sets Gy and G0y such that x ∈ G0y and y ∈ Gy . Clearly F ⊂ y∈F Gy and since X is λ -
S

compact, we can write F ⊂ y∈E Gy = A where, ET⊆ F and |E| < λ . Also we have x ∈ y∈E G0y = B.
S T

Since X is a Pλ -space, B is open and obviously A B = 0. /

Corollary 2.8. Every λ -compact Pλ -space is λ -normal.

3 Main result
Theorem 3.1. Let (X, τ) be a λ -compact pλ -space and (Y, τ 0 ) a λ -compact space. Then (X ×Y, τ ×
τ 0 ) is λ -compact.
Proof. Let U = {Tα ×Uα : α ∈ A} be an open cover of X ×Y . if x ∈ X, then for each y ∈ Y , there
exist Tα(x,y) ∈ τ and Uα(x,y) ∈ τ 0 such that (x, y) ∈ Tα(x,y) ×Uα(x,y) . Since U is an open cover of X ×Y ,
the family {{x} ×Uα(x,y) : Uα(x,y) ∈ τ 0 } is an open cover of {x} ×Y . Clearly, {x} ×Y is λ -compact
and therefore there exists a subcover {{x} ×Uα(x,yk ) : k ∈ K, |K| < λ } which covers {x} ×Y . Hence
the collection {Tα(x,yk ) ×Uα(x,yk ) : k ∈ K, |K| < λ } is a cover for {x} ×Y . Now put Tx = k∈K Tα(x,yk ) .
T

Since X is a pλ -space, Tx is an open subset in X. Thus {Tx × Uα(x,yk ) : k ∈ K, |K| < λ } is an open
cover of Tx × Y , a fortiori {Tα(x,yk ) × Uα(x,yk ) : k ∈ K} is an open cover of Tx × Y whose cardinality
is less than λ . Now since T = {Tx : x ∈ X} is an open cover for X, and X is λ -compact, there exists

3
 {Txs : s ∈ S, |S| < λ } ⊆ T for X. Therefore X × Y ⊆ ( s∈S Txs ) × Y = s∈S (Txs × Y ) ⊆
S S
aSsubcover
s∈S k∈K α(xs ,yk ) ×Uα(xs ,yk ) ). Since λ is regular and |S|, |K| < λ , the proof is complete.
S
(T

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