Some notes on topological calibers

We show that the definition of caliber given by Engelking in R. Engelking,"General topology", Sigma series in pure mathematics, Heldermann, vol. 6, 1989, which we will call caliber*, differs from the traditional notion of this concept in some cases and agrees in others. For instance, we show that if $\kappa$ is an infinite cardinal with $2^{\kappa}<\aleph_\kappa$ and $cf(\kappa)>\omega$, then there exists a compact Hausdorff space $X$ such that $o(X)=2^{\aleph_\kappa}=|X|$, $\aleph_\kappa$ is a caliber* for $X$ and $\aleph_\kappa$ is not a caliber for $X$. On the other hand, we obtain that if $\lambda$ is an infinite cardinal number, $X$ is a Hausdorff space with $|X|>1$, $\phi\in \{w ,nw\}$, $o(X) = 2^{\phi(X)}$ and $\mu := o\left(X^\lambda\right)$, then the calibers of $X^\lambda$ and the true calibers* (that is, those which are less than or equal to $\mu$) coincide, and are precisely those that have uncountable cofinality.


Introduction
The classical definition of caliber for a topological space states that if κ is an infinite cardinal, then κ is a caliber for X if for every collection {U α : α < κ} of non-empty open subsets of X, there exists J ⊆ κ such that|J| = κ and {U α : α ∈ J} satisfies α∈J U α = ∅ (see [13]).
However, despite the fact that this convention seems to be the most used (see, for example, [4], [9] and [15]), in [5, 2.7. 11, p. 116] Engelking defines: an infinite cardinal κ is a caliber for a topological space X if for every collection U of cardinality κ of non-empty open subsets of X, there exists a subcollection V ⊆ U , such that |V | = κ and V = ∅.
Naturally, an immediate observation is that in Šanin's definition repetitions in the enumeration are allowed; therefore, this is an indicator that these two notions of caliber may not coincide.The main goal of this paper is to delve into the differences that lie between these concepts and to expose that they indeed differ in some aspects.
We will call an infinite cardinal number κ caliber* if it satisfies the definition given by Engelking, and we will call it simply caliber if it satisfies Šanin's definition.
In Section 2 we list, without proof, some basic known results on calibers.Section 3 is devoted to obtaining basic results on calibers* and some of their variations such as precalibers* and weak precalibers*.Furthermore, in this section, we present examples of topological spaces for which their non trivial calibers* (those that do not exceed the number of open subsets) do not coincide with their calibers.Finally, we calculate calibers* in concrete classes of topological spaces like those hyperconnected and in βω.In Section 4, we study the calibers* for topological sums, and in Section 5 we analyze calibers* for topological products.

Preliminaries
Any topological or set-theoretic concept that is not explicitly defined in this text should be understood as in [5] and [10], respectively.
For a topological space X, the symbol τ X is the topology of X, and we will denote by τ + X the set τ X \ {∅}.For a cardinal number κ, we will use the symbol D(κ) to denote the discrete space of cardinality κ.
If α is an ordinal number, ℵ α will represent the cardinal number of the set ω α (see [10]).Furthermore, we will use the symbol ω to denote the first infinite ordinal and, therefore, the first infinite cardinal.The cardinal number of R, the set of real numbers, will be denoted by c.
If X is a set, the expression "X is a countable set" will mean that there exists an injective function from X into ω.If κ is a cardinal number, we will use the symbol [X] <κ to refer to the collection of subsets of X of size (cardinality) less than κ.In the same way, [X] ≤κ will represent the family of all subsets of size at most κ, and we will denote by [X] κ the set [X] ≤κ \ [X] <κ .Finally, P(X) will be the power set of X.
The cofinality of a cardinal κ, cf(κ), is the minimum ordinal number for which there exists a function f : cf(κ) → κ so that for every α < κ, there exists β < cf(κ) with α < f (β).Naturally, we always have cf(κ) ≤ κ.We will say that an infinite cardinal κ is regular if κ = cf(κ); otherwise, we say that κ is singular.Lastly, we will denote by κ + the successor cardinal of κ; that is, the first cardinal λ that satisfies the relation κ < λ.
Throughout this work we will denote by CN the proper class formed by the infinite cardinal numbers.For a topological space X, if U is a pairwise disjoint collection of τ + X , we say that U is a cellular family in X.The terms centered family and family with the finite intersection property in X mean the same; that is, they are used to designate a collection of subsets such that the intersection of every nonempty finite subcollection is non-empty.We say that a subset U of τ + X is linked if we have U ∩ V = ∅ for any U, V ∈ U .
We will be working throughout this paper with the following collections of cardinal numbers: WP(X) := {κ ∈ CN : κ is a weak precaliber for X}; P(X) := {κ ∈ CN : κ is a precaliber for X}; and C(X) := {κ ∈ CN : κ is a caliber for X}.
Our basic reference texts for material related to topological cardinal functions will be [8] and [9].However, for the purposes of this text, the symbols o(X) and |X| will represent, correspondingly, the cardinality of τ X and the cardinality of X; that is, we will dispense adding ω to these two cardinal numbers.
We compile below some known results that will be useful in the development of this text.
Lemma 2.3.If X is a topological space, κ is a weak precaliber for X, and U is a cellular family in X, then |U | < κ.Proposition 2.4.If X is a topological space, D is a dense subset of X, and κ is an infinite cardinal, then the following statements are true.
(1) κ is a precaliber (resp., weak precaliber) for D if and only if κ is a precaliber (resp., weak precaliber) for X. (2) If κ is a caliber for D, then κ is a caliber for X.

Chain* conditions
A cardinal number κ is a caliber* (resp., precaliber*; weak precaliber*) of the topological space X if for every family U of cardinality κ of non-empty open subsets of X, there exists a subcollection V ⊆ U , such that |V | = κ and V has non-empty intersection (resp., is centered; is linked).In the remainder of this text we will be working with the following collections: WP * (X) := {κ ∈ CN : κ is a weak precaliber* for X}; P * (X) := {κ ∈ CN : κ is a precaliber* for X}; and C * (X) := {κ ∈ CN : κ is a caliber* for X}.
In this section we will present several basic results regarding the relations between the chain conditions that we have presented.Let us start with the following result.Lemma 3.1.Let X be a topological space and κ a regular cardinal.Then κ is a caliber (resp., precaliber; weak precaliber) for X if and only if κ is a caliber* (resp., precaliber*; weak precaliber*).
Proof.We give a proof for the case of caliber.The other two cases can be similarly proved.Note that the direct implication is immediate and does not need the regularity of κ; it is sufficient to enumerate a U ∈ [τ + X ] κ without repetitions.To verify the converse implication, suppose that {U α : α < κ} is a subset of τ + X and define f : κ → {U α : α < κ} by f (α) := U α .If {U α : α < κ} < κ, then the regularity of κ guarantees that there exists β < κ such that f κ because the function g : J → V given by g(α) := U α is surjective, and {U α : α ∈ J} = ∅.
An equivalence of the chain conditions in terms of indexed families will help.Lemma 3.2.If X is a topological space and κ is a cardinal number, then the following statements are equivalent.
In what follows we will constantly use the equivalence exposed in Lemma 3.2, even without making explicit reference to it.
Let us fix a topological space X.The inclusions C(X) ⊆ P(X) ⊆ WP(X) and C * (X) ⊆ P * (X) ⊆ WP * (X) are easily deduced from the definitions.Furthermore, the direct implication of Lemma 3.1 indicates that C(X) ⊆ C * (X), P(X) ⊆ P * (X) and WP(X) ⊆ WP * (X).On the other hand, if R is the subclass of CN formed by the regular cardinals, Lemma 3.1 ensures that C * (X) ∩ R ⊆ C(X), P * (X) ∩ R ⊆ P(X) and P * (X) ∩ R ⊆ P(X).These lines are the proof of the following result.Proposition 3.3.For a topological space X the following inclusions hold.
(2) C(X) ⊆ C * (X), P(X) ⊆ P * (X) and WP(X) ⊆ WP * (X). ( The relations exposed in Proposition 3.3 can be displayed in a more friendly way in the following diagram (the relation A → B means A ⊆ B): The question that we would now like to answer is: Is it true that the inclusions C * (X) ⊆ C(X), P * (X) ⊆ P(X) and WP * (X) ⊆ WP(X) also hold?This question can be answered with the help of the following result.Proposition 3.4.If X is a topological space and κ is an infinite cardinal such that o(X) < κ, then κ ∈ C * (X).
Proof.If κ were not a caliber*, then there would exist a family U of cardinality κ of open subsets of X such that no subcollection of U of cardinality κ has a non-empty intersection, but there can be no such family U since o(X) < κ.
We will see in Theorems 3.11 and 3.20 that, consistently, the answer to Question 3.5 is in the negative.To do this we first need to prove a couple of auxiliary results.
Proof.For each α < κ, consider the set Suppose further that o(U α ) ≥ ℵ α , provided that α < κ.Let us fix a family {U (α, γ) : γ < ℵ α } ⊆ τ U + α , enumerated without repetitions, for every α < κ.For each ordinal γ < ℵ κ define V γ := U (α, γ) if and only if γ ∈ I α .Under these circumstances, it is not difficult to confirm that the family Although the notions of caliber and precalibers do not always coincide with their * versions, they do have some similarities.For example, our following result is the natural generalization of Lemma 2.3 for weak precaliber*.Lemma 3.7.If X is a topological space, κ is a weak precaliber* for X and U is a cellular family in X, then |U | < κ.
Proof.Let U be a subset of τ + X such that |U | ≥ κ.If we take V in [U ] κ , we can use that X has weak precaliber* κ to find a linked family W ∈ [V ] κ .In particular, U cannot be a cellular family.
A natural question suggested by Lemma 3.7 is whether the converse of this result is also valid; that is, is it true that if X is a topological space, κ is a cardinal number and any cellular family in X has cardinality less than κ, then κ is a weak precaliber* for X? To answer this question in the affirmative in certain particular cases, we first need to carry out a brief combinatory interlude.
If κ, λ, µ and ν are cardinal numbers, then κ → (λ) µ ν will mean, as usual, that for any set X with |X| = κ and any function f : A well known combinatory result is Ramsey's Theorem (see [6, Theorem 10.2, p. 66]): It is in this class of cardinals that we can give a converse of Lemma 3.7.The authors thank Jorge Antonio Cruz Chapital for suggesting the proof that we will present below.
Proposition 3.9.If X is a topological space and κ = ω or κ is weakly compact, then the following statements are equivalent.
(1) κ is a weak precaliber* for X. ( Proof.The implication (1) → (2) follows from Lemma 3.7.To verify the converse suppose that κ is not a weak precaliber* for X, and let U Finally, since V is not linked, f cannot be the constant 1 in the set [V ] 2 , and therefore we deduce that V is a cellular family in X of cardinality κ.
Corollary 3.10.If X is an infinite topological space and ω is a weak precaliber* for X, then X does not have the Hausdorff property.
To verify that ℵ κ ∈ C * (X), suppose that U is an element of [τ + X ] ℵκ and let {U α : α < ℵ κ } be an enumeration without repetitions of U .For each α < ℵ κ , there exist ) is injective, we obtain the following contradiction: Therefore, there exists I ∈ [ℵ κ ] ℵκ such that ∅ ∈ {W α : α ∈ I} and W α = W β for any distinct α, β ∈ I.Under these circumstances, since Z has caliber ℵ κ (see Lemma 2.7), there exists J ∈ In Theorem 3.11 it might be tempting to consider the softer constraint 2 κ ≤ ℵ κ , however, since (2 κ ) κ = 2 κ and (ℵ κ ) κ > ℵ κ (see [10,Lemma 10.40 Moreover, since Theorem 3.11 only deals with cardinals with uncountable cofinality, it remains to find out if it is possible to achieve a similar result for uncountable cardinals with countable cofinality.We will settle this debt later in Theorem 4.8.
Since for any U ∈ τ X the inclusion τ + U ⊆ τ + X is satisfied, the following result is immediate.
It is easy to verify that the converse implication in the previous result is not true.Indeed, if κ is an infinite cardinal with cf(κ) > ω, and we denote by X the free sum R ⊕ D(κ), then R ∈ τ X and κ is a caliber for R (see Lemma 2.2).However, since D(κ) admits a cellular family of cardinality κ, then Lemma 3.7 guarantees that D(κ) has no weak precaliber* κ.Proposition 3.12 ensures that κ is not a weak precaliber* for X.
Recall that a Hausdorff space is H-closed if it is closed in any of its Hausdorff extensions; for example, any compact Hausdorff space is H-closed.Naturally, a space is locally H-closed if any point admits an H-closed neighborhood.A classical Hausdorff extensions result guarantees that if X is a locally H-closed Hausdorff space, then X is open in any of its Hausdorff extensions (see [11,Proposition (b), p .543]).
Proof.The three equalities follow from Propositions 2.4 and 2.5, while the three inclusions are a consequence of Proposition 3.12.
Compare the following result with Proposition 2.4(2).Proposition 3.14.If X is a topological space, D is a dense subset of X, κ is an infinite cardinal, and κ is a precaliber* (resp., weak precaliber*) for X, then κ is a precaliber* (resp., weak precaliber*) for D.
The corresponding result for calibers* is invalid.Indeed, Lemma 2.7 guarantees that if κ has uncountable cofinality, then D(2) κ has caliber κ.On the other hand, a routine argument shows that Consequently, κ is not a caliber* for D. Question 3.15.Is it true that κ is a caliber* (resp., precaliber*; weak precaliber*) for a topological space X, if κ is a caliber* (resp., precaliber*; weak precaliber*) for a dense subset D of X?
We will show in Corollaries 3.23 and 4.11 that the answer to Question 3.15 is in the negative.However, we will first discuss some simple cases in which the answer is in the affirmative.For instance, the following result is a consequence of Proposition 2.4 and Lemma 3.1.Proposition 3.16.If X is a topological space, D a dense subset of X, and κ a regular cardinal that is a caliber* (resp., precaliber*; weak precaliber*) for D, then κ is a caliber* (resp., precaliber*; weak precaliber*) for X.
Corollary 3.19 below shows that in a certain class of extensions the answer to Question 3.15 is in the affirmative.The following result is a consequence of Proposition 4.4 and Theorem 4.5 that we will prove later.Proposition 3.17.Let X be a topological space and Y an extension of X.If κ is a caliber* (resp., precaliber*; weak precaliber*) for X and for the remainder Y \ X, then κ is a caliber* (resp., precaliber*; weak precaliber*) for Y .
Moreover, it is possible to give a kind of reciprocal result for the above by virtue of Proposition 3.12.Proposition 3.18.Let X be a topological space and Y an extension of X.If κ is a caliber* (resp., precaliber*; weak precaliber*) for Y , and Y \ X is closed in Y , then κ is a caliber* (resp., precaliber*; weak precaliber*) for X.
With the previous result available it is possible to strengthen Theorem 3.11 as follows: Proof.Let X be the one-point compactification of the space whose existence is guaranteed by Theorem 3.11.Clearly, X is a compact Hausdorff space that satisfies the relations o(X) = 2 ℵκ = |X|.Furthermore, Proposition 2.4(1) confirms that ℵ κ ∈ WP(X), while Corollary 3.19 ensures that ℵ κ ∈ C * (X) and κ ∈ WP * (X).Question 3.21.Is it possible to construct in ZFC an example of an infinite Hausdorff space X and an infinite cardinal κ such that κ ∈ C * (X) and κ ∈ C(X)?
In order to give a negative answer to Question 3.15, let us first prove the following theorem.
Theorem 3.22.If κ is a singular cardinal and X is a topological space such that cf(κ) is not a caliber* (resp., precaliber*; weak precaliber*) for X, then there exists an extension Y of X such that κ is not a caliber* (resp., precaliber*; weak precaliber*) for Y .
Corollary 3.23.If κ is a regular uncountable cardinal with 2 κ < ℵ κ , then there exist a topological space X and an extension Y of X such that ℵ κ ∈ C * (X) and ℵ κ ∈ WP * (Y ).
Proof.Let X be as in Theorem 3.11.Since ℵ κ is a singular cardinal and cf (ℵ κ ) = κ is not a weak precaliber* for X, Theorem 3.22 produces an extension Y of X such that ℵ κ is not a weak precaliber* for Y .Thus, ℵ κ ∈ C * (X) and ℵ κ ∈ WP * (Y ).
In general, it is not a simple task to determine precisely what the chain conditions for topological spaces are.In what follows we will try to exemplify this fact by means of some well known spaces.
Recall that an infinite family A formed by infinite subsets of ω is almost disjoint if |A ∩ B| < ω, provided that A, B ∈ A are different.A classical result guarantees the existence of an almost disjoint family of cardinality c (see, for example, [10, Theorem 1.3, p. 48]).In particular, for any cardinal κ with ω ≤ κ ≤ c, there exists an almost disjoint family of cardinality κ on ω.Claim.The collection {A α : α < κ} is a faithfully indexed almost disjoint family such that, for any J ∈ [κ] κ , there exist α, β, γ ∈ J with Let us first notice that, if α < β < κ, and n, m ∈ ω are such that α ∈ I n and β ∈ I m , then the relation , which is absurd since B α is infinite, while this last union is finite.For this reason, A α = A β .Also, since |A α ∩A β | ≤ |B α ∩B β | < ω, and {A γ : γ < κ} is clearly formed by infinite subsets of ω, it follows that {A γ : γ < κ} is an almost disjoint family enumerated without repetitions.
On the other hand, if J ∈ [κ] κ and α, β ∈ J are different, there exists m < ω with A α ∩A β ⊆ m.Thus, if we take n > m with I n ∩J = ∅ and γ ∈ I n ∩J, then the relations Finally, the Claim guarantees that A := {A α : α < κ} is the family required in the statement of the present lemma.
In particular, certain chain conditions depend entirely on how the cardinal numbers are related to each other.
Corollary 3.29.The following statements are true.
For our next example it is convenient to establish a couple of auxiliary results.Recall that a topological space X is hyperconnected if any two non-empty open subsets of X intersect.In other words, X is hyperconnected if and only if τ + X is a linked family.It turns out that this condition is equivalent to a property that is seemingly stronger.

Proposition 3.32. A topological space is hyperconnected if and only if τ +
X is a centered family.
Proof.The inverse implication is clear.To verify the direct implication, we proceed by finite induction.Suppose that for an n < ω it has already been shown that the intersection of any n elements of τ + X is a non-empty set.Now suppose that {U k : k < n + 1} is a subset of τ + X of size n + 1.It then follows that {U k : k < n} is an element of τ + X and therefore, {U k : k < n + 1} is non-empty.From the previous proposition, two more conditions equivalent to being hyperconnected can be obtained in terms of precalibers and weak precalibers.For the following result only, we naturally extend the concepts of precaliber, precaliber*, finite k with 2 ≤ k < o(X) is a weak precaliber* for X, then X is a hyperconnected space.Now let us assume that 3 is a weak precaliber* for a space X.When 3 < o(X), the previous paragraph implies that X is hyperconnected; otherwise, the condition o(X) ≤ 3 implies that either τ X = {∅, X} or there exists A ∈ P (X) \ {∅, X} such that τ X = {∅, X, A}.In either case the space X is hyperconnected.
The classical examples of hyperconnected spaces are the infinite spaces equipped with the cofinite topology.What we will do next is calculate all the above chain conditions for this class of spaces.Recall that an infinite space X is called cofinite if τ X = {∅} ∪ {U ⊆ X : X \ U < ω}.The following observation is essential.
Remark 3.34.If X is a cofinite space and κ is a cardinal number, then κ is a caliber for X if and only if for any {F α : α < κ} ⊆ [X] <ω there exists J ∈ [κ] κ such that {F α : α ∈ J} = X.Lemma 3.35.If X is the cofinite space of cardinality κ, then: (1) Proof.First, since X is a hyperconnected topological space, Proposition 3.33 ensures that CN ⊆ P(X).Consequently, item (1) is true.Item ( 2) is a consequence of the equality o(X) = κ and Proposition 3.4.
The above result allows us to fully classify the chain conditions on cofinite spaces.
Proof.Clearly, it is sufficient to verify that CN ⊆ C(X).With this objective in mind, we will first argue that κ ∈ C(X).
A natural question is whether in Theorem 3.39 the equality C * (X) = {λ ∈ CN : λ > ω} holds.We close this section with a theorem in which we show that this relation is independent of ZFC.Recall that the Continuum Hypothesis, CH, is the equality "c = ω 1 ".Theorem 3.40.The following statements hold.
Proof.To verify item (1) let X be a countably infinite T 1 space.An immediate observation is that o(X) ≤ c since τ X is a family of subsets of X.Thus, under CH it is satisfied that o(X) ≤ ω 1 .Finally, use the relations For item (2) let X be the set ω equipped with the discrete topology.Use Lemma 3.24 to find an almost disjoint family A such that |A | = ℵ ω and, for any X of size ℵ ω , we deduce that ℵ ω is not a precaliber* for X.Consequently, C * (X) = {λ ∈ CN : λ > ω}.

Chain conditions in topological sums
For the purposes of this section, λ will always represent a positive cardinal number, not necessarily infinite.The following result is a simple generalization of [16, T.276, p. 311].
With Theorem 4.6 available, we can obtain within ZFC a result similar to Theorems 3.11 and 3.20 for uncountable cardinals with countable cofinality.
With an additional hypothesis, we can obtain a variant of Theorem 4.8 in which κ ∈ WP(X) is achieved.Theorem 4.9.If κ is an uncountable cardinal with κ > c and cf(κ) = ω, then there is a T 1 locally compact space X such that o(X) = κ = |X| and κ ∈ C * (X) \ WP(X).
Proof.Let Y be the cofinite space of size κ, Z the discrete space of size ω, and set X := Y ⊕ Z. Evidently, X is locally compact, T 1 and o(X) = κ = |X|.Now, since in Example 3.31 we proved that ω is not a weak precaliber for Z, Proposition 2.5 implies that ω ∈ WP(X) and thus, κ ∈ WP(X) (see Lemma 2.1).Finally, by Propositions 3.4 and 3.37, and Theorem 4.6, we conclude that κ ∈ C * (X).
Note that if ω ≤ κ ≤ λ is a cardinal number, then clearly κ is not a weak precaliber* for {X α : α < λ}, even if κ is a caliber for every X α .In our following example we will show that the condition cf(κ) > λ is essential in Theorem 4.7, even when κ > λ.
We proved in Theorem 4.8 that if X stands for the topological sum of the cofinite space of size ℵ ω with the cofinite space of size ω, then ℵ ω ∈ C * (X).In this particular example, it is a consequence of Propositions 3.12 and 3.36 that ω ∈ C * (X).For this reason, Theorem 3.22 implies the following corollary.
Corollary 4.11.There is a topological space X and an extension Y of X such that ℵ ω ∈ C * (X) and ℵ ω ∈ C * (Y ).

Chain conditions in topological products
There are several results about the preservation of the classical chain conditions for topological products in the literature (see, for example, [1], [12], [13] and [15]).In this section we will present some preservation results related to chain* conditions.Lemma 5.1.Let f : X → Y be a continuous surjective function.If κ is a caliber* (resp., precaliber*; weak precaliber*) for X, then κ is a caliber* (resp., precaliber*; weak precaliber*) for Y .
In what follows we will show that chain conditions and chain* conditions "truly" coincide on Cantor cubes, i.e., if κ is a caliber* for D(2) λ with κ ≤ 2 λ , then κ is a caliber for D(2) λ .Theorem 5.3.Let κ be a cardinal number with cf(κ) = ω and Z a topological space with o(Z) ≥ κ.If X ∼ = Y × Z, where Y is an infinite Hausdorff space, then X does not have weak precaliber* κ.
For every n < ω define W n := {V n × U α : κ n ≤ α < κ n+1 }.Then W := n<ω W n is a family of κ non-empty open subsets of Y × Z.We claim that for every subset W ′ ⊆ W of size κ, we can find two distinct elements of W ′ with empty intersection.Indeed, since |W ′ | = κ, there are n < m < ω such that W ′ contains an element of W n and an element of W m , say For every infinite cardinal λ, D(2) λ ∼ = D(2) λ × D(2) λ , therefore we obtain: Corollary 5.4.If κ is a cardinal number with κ ≤ 2 λ and cf(κ) = ω, then D(2) λ does not have weak precaliber* κ.
Corollary 5.5.If λ is a cardinal number, then: (1) C D(2 Proof.To verify the first equalities, it is enough to observe that if κ is a cardinal number with cf(κ) > ω, then Lemma 2.7 implies that κ is a caliber for D(2) λ .On the other hand, since D(2) λ is an infinite Hausdorff space, no infinite cardinal with countable cofinality can be a weak precaliber for D(2) λ (see Lemmas 2.1 and 2.3).
For the second part, the relations κ ∈ CN : cf(κ) > ω = C D(2) λ ⊆ C * D(2) λ together with Lemma 2.8 and Proposition 3.4 imply that Lastly, Corollary 5.4 ensures that no κ ≤ 2 λ with countable cofinality is a weak precaliber* for D(2) λ .We can extend Corollary 5.5 for certain classes of infinite Hausdorff spaces.In order to do this we first need to prove some auxiliary results regarding the behavior of the cardinal function o in topological products.Let us consider the following question.
Question 5.6.Let X be a topological space and λ be a cardinal number.Under what conditions on X and λ is it possible to express o(X λ ) in terms of λ and φ(X) for some topological cardinal function φ?In particular, under what conditions on From now on, the phrase "φ is a topological cardinal function" means that φ is a correspondence rule that assigns to each topological space X an infinite cardinal number φ(X) such that, if X is homeomorphic to Y , then φ(X) = φ(Y ).
Let P be a class of topological spaces and φ be a topological cardinal function.We will say that (P, φ) is a finitely productive pair if P is closed under finite products and, for each X, Y ∈ P, o(X) ≤ 2 φ(X) and φ(X × Y ) = φ(X) • φ(Y ).Similarly, (P, φ) will be called a productive pair if P is closed under arbitrary products and, for every Y ∈ P and {Y α : α < λ} ⊆ P, o(Y ) ≤ 2 φ(Y ) and φ α<λ Y α = λ • sup φ(Y α ) : α < λ .These conventions will allow us to formulate with more simplicity the results that we will present below.Proposition 5.8.Let (P, φ) be a finitely productive pair.If X, Y ∈ Corollary 5.9.Let φ ∈ {w, nw} (see [8]).If X and Y are topological spaces such that o Let X and Y be a pair of topological spaces.As already noted, the function For this reason, under GCH the answer to Question 5.10 is affirmative since, with this hypothesis, o(X × Y ) ∈ {µ, 2 µ }.
The following is a generalization of Proposition 5.8.
It is known that if there are no inaccessible cardinals and GCH is true, then any infinite Hausdorff space satisfies o(X) = 2 κ for some cardinal κ (see [8,Theorem 13.1,p. 48]).This observation combined with what we have discussed so far suggests the following question: Question 5.13.If X is a topological space and φ is a cardinal function, under what conditions is the equality o(X) = 2 φ(X) satisfied?For example, regarding the cardinal function φ = w, in [8, Theorem 8.1(e), p. 32] it is shown that, if X is infinite and metrizable, then o(X) = 2 w(X) .On the other hand, in the realm of compact Hausdorff spaces it is not always satisfied that o(X) = 2 w(X) .To illustrate this fact notice that, if X is the Alexandroff-Urysohn double arrow (see [8,Example 14.4, p. 51]), then o(X) = c = w(X).However, in certain subclasses of compact T 2 spaces we can obtain the relation o(X) = 2 w(X) .
A space is extremely disconnected if every open subset of it has an open closure.In [2, Corollary 1, p. 608] it is proved that extremely disconnected compact spaces satisfy the equality |X| = 2 w(X) .Consequently, since |X| ≤ o(X) ≤ 2 w(X) for T 1 -spaces, we conclude that o(X) = 2 w(X) .For example, any Stone space of a complete Boolean algebra verifies this relationship (see [11,Theorem (d), p. 448]).
On the other hand, we say that a compact space is polyadic if it is a continuous image of a power of the one-point compactification of an infinite discrete space.If X is a polyadic compact space, then X admits a discrete subspace of cardinality w(X) (see [7,Corollary 1,p. 15]).Thus, since for each discrete subspace D ⊆ X it is satisfied that 2 |D| ≤ o(X), we deduce the equality o(X) = 2 w(X) .
Back to our original goal, with this background we are better positioned to generalize Corollary 5.5.

It turns out that o
α∈λ\J X α = o(X) (see Lemma 5.7) and thus, Lemma 5.14 implies that X does not have weak precaliber* κ.
In [15] it was shown that, if κ is an infinite cardinal, X is a topological space with caliber κ, cf(κ) > ω and λ is a cardinal number, then the power X λ also has

Lemma 3 .
24.If κ ≥ ω is a cardinal number with κ < c and cf(κ) = ω, then there exists an almost disjoint family A on ω, such that |A | = κ and for any B ∈ [A ] κ , there exist A, B, C ∈ B with A ∩ B ∩ C = ∅.Proof.Let {κ n : n < ω} be a strictly increasing sequence of cardinal numbers with κ 0 = 0 and sup{κ n : n < ω} = κ.Also, for each n < ω, let us define I n := [κ n , κ n+1 ).Furthermore, let us use the relation κ < c to fix an almost disjoint family {B α : α < κ} enumerated without repetitions.The last step is to define for any α < κ, A α := B α \ n if and only if α ∈ I n .

Remark 3 .
25.If A is an almost disjoint family on ω and U ∈ βω \ ω, then |A ∩ U | ≤ 1.Indeed, if A, B ∈ A are different, then |A ∩ B| < ω and thus, since U is free, we deduce that A ∩ B ∈ U ; consequently, {A, B} ⊆ U .Lemma 3.26.If κ ≥ ω is a cardinal number with κ < c and cf(κ) = ω, then κ ∈ P * (βω).Proof.Let A be an almost disjoint family with the characteristics of Lemma 3.24, and consider the family A := {A * : A ∈ A }. Since each element of A is contained in a free ultrafilter on ω, Remark 3.25 implies that A ∈ [τ + βω ] κ .Now let B ∈ [A] κ , B ∈ [A ] κ where B = {B * : B ∈ B}, and A, B, C ∈ B such that A ∩ B ∩ C = ∅.Thus, there is no U ∈ βω such that {A, B, C} ⊆ U ; that is, A * ∩ B * ∩ C * = ∅.Hence, A is an element of [τ + βω ] κ and, for any B ∈ [A] κ , B is not a centered family.Consequently, κ is not a precaliber* for βω.Theorem 3.27.The following statements are true.
the same conclusion is obtained when X and Y are infinite and φ = |•|.Proposition 5.8 suggests the following natural question: Question 5.10.Is it true that if X satisfies o(X) = 2 κ and Y satisfies o(Y ) = 2 λ , then o(X × Y ) is of the form 2 θ for some cardinal θ?