Weakly Separated Spaces and Pixley–Roy Hyperspaces

In this paper we obtain new results regarding the chain conditions in the Pixley–Roy hyperspaces F[X]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {F}\hspace{0mm}}[X]$$\end{document}. For example, if c(X) and R(X) denote the cellularity and weak separation number of X (see Sect. 4) and we define the cardinals c∗(X):=sup{c(Xn):n∈N}andR∗(X):=sup{R(Xn):n∈N},\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} c^* (X):= \sup \{c(X^{n}): n\in {\mathbb {N}}\} \quad \text {and} \quad R^{*}(X):= \sup \{R(X^{n}): n\in {\mathbb {N}}\}, \end{aligned}$$\end{document}then we show that R∗(X)=c∗F[X]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R^{*}(X) = c^ {*}\left( {\mathscr {F}\hspace{0mm}}[X]\right) $$\end{document}. On the other hand, in Sakai (Topol Appl 159:3080–3088, 2012, Question 3.23, p. 3087) Sakai asked whether the fact that F[X]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {F}\hspace{0mm}}[X]$$\end{document} is weakly Lindelöf implies that X is hereditarily separable and proved that if X is countably tight then the previous question has an affirmative answer. We shall expand Sakai’s result by proving that if F[X]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {F}\hspace{0mm}}[X]$$\end{document} is weakly Lindelöf and X satisfies any of the following conditions: X is a Hausdorff k-space; X is a countably tight T1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_1$$\end{document}-space; X is weakly separated, then X is hereditarily separable. X is a Hausdorff k-space; X is a countably tight T1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_1$$\end{document}-space; X is weakly separated,


Introduction
Chain conditions on topological spaces have been studied after their introduction by Šanin in [19].Since then, many research articles have been published on the behavior of calibers, precalibers, and weak precalibers in topological spaces.For example, in [14], [19] and [21] some results can be found regarding the preservation of these notions in the realm of topological products.
On the other hand, the Pixley-Roy hyperspaces F [X] have also been the focus of plentiful research since their presentation by Pixley and Roy in [13].Regarding these spaces, numerous papers and surveys have also been written (see, for example, [2] and [10]).
It was not until Sakai's article [17] that the behavior of the "precaliber ω 1 " notion in Pixley-Roy hyperspaces was studied for the first time.In that article some very interesting connections were made regarding weakly separated spaces and other cardinal functions of the hyperspace F [X].
In the present work we will delve into the study of cellularity, calibers, precalibers and weak precalibers in the hyperspaces F [X] and their relationship with other various topological concepts (ω-covers, the weak separation number, the tightness, discrete families, k-spaces, etc).

Preliminaries
All topological and set-theoretic notions that are not explicitly mentioned in this paper should be understood as in [3] and [9], respectively.Throughout the text and unless explicitly stated otherwise, all spaces and cardinal numbers considered will be infinite.
The symbol ω will stand for both, the set of all non-negative integers and the first infinite cardinal.Additionally, the symbol N will stand for the set ω \ {0}.If κ is a cardinal number, the cofinality of κ will be denoted by cf(κ).On the other hand, if X is a set, the symbols [X] <ω and [X] κ will represent the families {Y ⊆ X : |Y | < ω} and {Y ⊆ X : |Y | = κ}, respectively.Furthermore, we will denote by CN the proper class formed by the cardinal numbers, and UC will stand for the subclass of CN made up by the cardinals with uncountable cofinality.
For a topological space X, we will use the symbol τ X to refer to the family of open subsets of X.Similarly, τ + X shall be used to denote the set τ X \ {∅} and τ * X will stand for the set {A ⊆ X : X \ A ∈ τ X }.Now, if U is a pairwise disjoint collection of τ + X , we will say that U is a cellular family in X.The term centered family shall be used to designate a collection of subsets such that the intersection of the elements of any of its non-empty finite subcollections is non-empty.On the other hand, we will 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}.
In the remainder of this portion of the text we will mention some auxiliary and basic propositions that we are going to use multiple times throughout the paper.Let X be a topological space and κ a cardinal number.Recall that d(X) stands for the density of X.
The following result can be found in [15,Theorem 3.37,p. 14].
The family {[F, U ] : F ∈ F (X) ∧ U ∈ τ X } is a basis for a topology on F [X] known as the Pixley-Roy topology.To follow the traditional notation of the literature, the symbol F [X] will stand for the set F (X) equipped with the Pixley-Roy.These spaces were introduced in 1969 by Pixley and Roy in [13] with the aim of presenting a non-separable Moore space with countable cellularity.It is well known that if X is a T 1 -space, then F [X] is zero-dimensional (hence, completely regular), Hausdorff and hereditarily metacompact (see [2]).
We are interested in determining precisely who the collections C F [X] , P F [X] , and WP F [X] .Notice that when X is finite then so is F [X]; thus, all infinite cardinals are calibers for F [X].For this reason, we must enforce the constraint |X| ∈ CN.Also, we will require X to be a T 1 -space so that the F [X] has nice separation properties.
With these conventions in mind, let us start by recalling a basic result regarding the density and cellularity of F [X] (see [18,Theorem 2,p. 337]).
It is appropriate to mention that in Proposition 3.1 equality is not necessarily reached in the relation hd(X) • hL(X) ≤ c F [X] .For example, if we denote the Sorgenfrey line by S and for each x ∈ S we define U x := [x, ∞), then it is easy to check that the collection [{x}, U x ] : x ∈ S is a cellular family in F [S] of size c, while hd(S) • hL(S) = ω.Now, the F [X] hyperspaces have a particular feature in that their collection of calibers can be easily determined, but their families of precalibers and weak precalibers are hard to find.Let us first focus on the collection C F [X] .
Proof.Combining Propositions 2.1 and 3.1 we immediately deduce that {κ ∈ CN : F would be a finite subset of X containing the infinite set {x α : α ∈ J}, which is absurd.This shows that cf(κ) is not a caliber for F [X] and hence we deduce from Proposition 2.3 that κ is also not a caliber for Thus, a combination of Proposition 2.4 with Theorem 3.3 implies that if X is a topological space, then In particular, Theorem 2.5 and the relations in (3.1) imply the following result.
Corollary 3.4.If X is a countably infinite space, then Furthermore, it is well known that many cardinal functions of F [X] coincide with |X| (see [18,Theorem 2,p. 337]).In the same vein as these results, if we define the Šanin number of X as the cardinal š(X) := min κ ∈ CN : κ + is a caliber for X , then Theorem 3.3 implies that š F [X] = |X|, in other words, the cardinal number š F [X] also enters the list of cardinal functions of F [X] that match the cardinality of X.
To calculate precisely who the collections P F [X] and WP F [X] are, we present the following definition.Definition 3.5.If X is a topological space and κ is a cardinal number, we say that X satisfies the condition C(κ) if and only if for any {x α : α < κ} ⊆ X and Remark 3.6.Every topological space that satisfies the condition C(κ) necessarily has caliber κ.Furthermore, it is clear that if Y is a subspace of X and X satisfies the property C(κ), then Y does too; that is, the property C(κ) is hereditary.
We will show later in Theorem 3.11 what is the relationship between C(κ) and the chain conditions of Pixley-Roy hyperspaces.What follows is to establish general properties about the C(κ) condition that we will use in the rest of this section.We will now show some results regarding the preservation of the condition C(κ) in certain topological constructions.
Before we begin, it is worth mentioning two things: first, the behavior of the C(κ) property is similar to that of usual chain conditions; and second, from Lemma 3.7 to Proposition 3.10 it is not necessary that the spaces considered be T 1 .
The following lemma can be proven by taking preimages and choosing points adequately.
Lemma 3.7.Let κ be a cardinal number and f : X → Y a continuous surjective function between topological spaces.If X satisfies C(κ), then so does Y .Proposition 3.8.Let κ be a cardinal number and X a topological space.If X satisfies C(κ), then X also satisfies C(cf(κ)).
Proof.The reciprocal implication follows from Lemma 3.9.To check the direct implication, suppose that X := {X α : α < λ} satisfies C(κ).Since each X α is a subspace of X, Remark 3.6 implies that X α satisfies C(κ).On the other hand, if we assume that cf(κ) ≤ λ, then cf(κ) is not a weak precaliber for X since {X α : α < cf(κ)} is a cellular family and thus, Proposition 2.3 ensures that κ is also not a weak precaliber for X, a contradiction to our hypothesis (see Remark 3.6).In sum, cf(κ) > λ.
In Section 5 we will determine the interaction between the condition C(κ) and the topological products.For now, it is a good idea to start making connections between the C(κ) property and Pixley-Roy hyperspaces with the results we have shown so far.
In [17, Theorem 3.18, p. 3087] Sakai proved that if X is a topological space, then F [X] has precaliber ω 1 if and only if X satisfies C(ω 1 ).As we show below, Sakai's argument can be adapted for any cardinal number with uncountable cofinality to obtain a fundamental equivalence for this section of the text.Theorem 3.11.If X is a topological space and κ is a cardinal number with cf(κ) > ω, then the following statements are equivalent.
Recursively construct a collection {J k : k < n} such that the following conditions are satisfied for any k < n: Remark 3.12.A consequence of Remark 3.6 is that if a space X satisfies the condition C(κ), then X has caliber κ hereditarily.For this reason, Theorem 3.11 implies that the last inclusion of (3.1) can be strengthened to However, in general, this relation is not an equality.For example, the Sorgenfrey line S is hereditarily separable (thus, it has hereditary caliber ω 1 ), but F [S] admits a cellular family of cardinality c (see Remark 3.2); in particular, F [S] does not have weak precaliber ω 1 .
To give us an idea of the strength of the condition C(ω 1 ), Sakai mentions in [17,Corollary 3.19,p. 3087] that if a space X satisfies C(ω 1 ), then necessarily X ω is hereditarily separable and hereditarily Lindelöf.
The last containment of (3.1) and Theorem 3.11 imply the following result.
In light of Corollary 3.13, our next objective is to try to find internal conditions to know when a space satisfies or not the condition C(κ) for a cardinal κ with uncountable cofinality.
First, we show below that when we consider the net weight of the space in a convenient way, we obtain a positive answer.Proposition 3.14.If X is a topological space and κ is a cardinal number such that nw(X) < cf(κ), then X satisfies C(κ).In particular, {κ ∈ Proof.Let N be a net for X of minimum cardinality, and {x α : α < κ} ⊆ X and An immediate consequence of Theorem 3.3 and Proposition 3.14 is that if X satisfies nw(X) < |X|, then the collections C F [X] and P F [X] do not match due to the relations In sum, the following corollary is verified.
Now, our next results are intended to set conditions on X to ensure that the condition C(κ) is not satisfied on X. Proposition 3.17.If X is a topological space and κ is a cardinal number such that κ ≤ d(X), then X does not satisfy C(κ).
Proof.The relation cf(κ) < hd(X) implies the existence of Y ⊆ X with cf(κ) < d(Y ).Then, by Proposition 3.17 it is verified that Y does not satisfy C(cf(κ)), and Proposition 3.8 ensures that Y does not satisfy C(κ).Lastly, since C(κ) is hereditary (see Remark 3.6), X does not satisfy C(κ).
Thanks to Proposition 3.14 and Corollary 3.18, the only cardinal numbers κ that remain to be analyzed are those that have uncountable cofinality and satisfy the relations To refine the first inequality of (3.2) it is necessary to further expand our conceptual field.

Weakly separated spaces
Unless explicitly stated otherwise, the topological spaces in this section will not be constrained to satisfy the separation axiom T 1 .
Recall that if X is a set, x ∈ X and < is an ordering relation on X, then the initial segment determined by x is the set (←, x) := {y ∈ X : y < x}.
A topological space X is right-separated (resp., left-separated) if it admits a well order such that its initial segments are elements of τ X (resp., of τ * X ).The height and width of X are, respectively, the cardinal numbers These types of spaces have been extensively studied in the literature, even in connection with the chain conditions of topological spaces (see [7] and [8]).A well known result is that hL = h and hd = z (see [7, 2.9, p. 16]).
On the other hand, a space X is weakly separated if there exists a family {U x : x ∈ X} ⊆ τ X that satisfies the following conditions: x ∈ U x for each x ∈ X, and if x, y ∈ X are distinct, then x ∈ U y or y ∈ U x .These spaces were introduced by Tkachenko in [23].In these circumstances we will say that the family {U x : x ∈ X} is a weak separation for X.Finally, the weak separation number of X is the cardinal number R(X) := sup{|Y | : Y is a weakly separated subspace of X} + ω.
It is easy to construct weakly separated spaces, e.g.any countable T 1 -space has this property.Indeed, if X is finite then {{x} : x ∈ X} is a weak separation for X (any discrete space is weakly separated).On the other hand, if {x n : n < ω} is an enumeration without repetitions of X and for each n < ω we define U n := X \ {x k : k < n}, then {U n : n < ω} is a weak separation for X.
One more example that is fundamental to us is the Sorgenfrey line.Just notice that the family {[x, ∞) : x ∈ S} is a weak separation for S.
Also, it is not difficult to check that all right-separated or left-separated subspaces of a space X are weakly separated; consequently, hd(X) • hL(X) ≤ R(X).Furthermore, as for any weakly separated space it is true that | • | = nw, then R(X) ≤ nw(X).We collect the observations of this paragraph in the following result.
Corollary 4.2.If nw(X) < |X|, then X is not weakly separated.In particular, uncountable cosmic spaces are not weakly separated.
We now expose some basic properties of weakly separated spaces.For example, routine arguments can be used to prove the following proposition.Proposition 4.3.Let X, Y and Z be a triplet of topological spaces.
(1) If Y is a subspace of X and Z is a subspace of Y , then Z is weakly separated as a subspace of X if and only if Z is weakly separated as a subspace of Y .(2) When X is weakly separated, X is T 0 .
(3) If X is weakly separated and Y is a subspace of X, then Y is weakly separated.(4) The following statements are true for a function f : X → Y .
(a) If f is a condensation and Y is weakly separated, then X is weakly separated.
Proposition 4.4.If X is a topological space and U is a family of weakly separated open subspaces of X, then U is a weakly separated subspace of X.
Proof.Let U := U , {U α : α < λ} be an enumeration without repetitions of U and, for each α < λ, fix a weak separation {V (x, α) : Proposition 4.5.Let X be a topological space and F a family of weakly separated closed subspaces of X.If F is locally finite with respect to F , then F is a weakly separated subspace of X.
Proof.Let F := F and {F α : α < λ} be an enumeration without repetitions of F .For each x ∈ F let U x ∈ τ F be such that x ∈ U x and I x := {α < λ : F α ∩U x = ∅} is a finite set, and define J x := {α ∈ I x : x ∈ F α } and K x := I x \ J x .Furthermore, for all α < λ let {V (x, α) : x ∈ F α } be a subset of τ F that weakly separates F α .Finally, for each x ∈ F define We will show {W x : x ∈ F } is a weak separation for F .Note that {W x : x ∈ F } is a subset of τ F with x ∈ W x for all x ∈ F .Now, let x, y ∈ F be distinct and α, β < λ be such that x ∈ F α and y ∈ F β .Suppose further that x ∈ W y and observe that, since x ∈ F α ∩ U y , α belongs to I y .Then, since for each γ ∈ K y we have that x ∈ X \ F γ , we infer that α ∈ K y and, therefore, α ∈ J y .Thus, x and y are elements of F α such that x ∈ V (y, α); consequently, since {V (z, α) : z ∈ F α } is a weak separation for F α , we obtain the relation y ∈ V (x, α), which implies that y ∈ W x .
The direct implication of the following result follows from Proposition 4.3(3), while the converse implication is a consequence of Proposition 4.4.
To check the remaining inequality, let Y be a weakly separated subspace of X.Since for every α < λ it is satisfied that Y ∩ X α is a weakly separated subspace of X α (see proposition 4.3), we deduce that |Y ∩ X α | ≤ R(X α ); thus Proposition 4.8.If λ is a cardinal number (not necessarily infinite) and {X α : α < λ} is a family of topological spaces with al least two points, then the following statements are true.
(2) Whenever each X α is weakly separated, the box product α<λ X α is weakly separated.In particular, the topological product of a finite family of weakly separated spaces is weakly separated.(3) α<λ X α is not weakly separated if λ ≥ ω.
Proof.For part (1) it is enough to remember that each X α is homeomorphic to a subspace of the topological product α<λ X α .
For part (3) notice that if J is a countably infinite subset of λ, for each α ∈ J we take x α , y α ∈ J distinct, and we define Y α := {x α , y α }, then the product α∈J Y α is cosmic, uncountable and embeds into α<λ X α .Therefore, since Corollary 4.2 guarantees that α∈J Y α is not weakly separated, we conclude that α<λ X α is also not weakly separated.
By virtue of Proposition 4.7, it would be desirable to obtain a similar formula to calculate the value of R for a product of topological spaces.We have not been able to obtain an equality in the previous sense, but we do have a couple of bounds in the case of T 2 -spaces.Proposition 4.9.If λ is a cardinal number (not necessarily infinite) and {X α : α < λ} is a family of Hausdorff spaces, then Proof.On the one hand, since α<λ X α embeds into α<λ X α , Proposition 4.7 implies that On the other hand, Proposition 4.1 guarantees that As our last basic property regarding weakly separated spaces, the following result is mentioned in [23].
In sum, Propositions 3.1 and 4.1 certify that for any space X the cardinal numbers R(X) and c F [X] are between hd(X) • hL(X) and nw(X).The natural question that arises is: how are the cardinals R(X) and c F [X] related?
Before answering the previous question, we are going to introduce a couple more cardinal functions.Following the tradition of the literature, for any topological space X we define c * (X) := sup{c(X n ) : n ∈ N} and R * (X) := sup{R(X n ) : n ∈ N}.
The connection between R(X), c F [X] , R * (X), and c * F [X] is established in Theorems 4.12, 4.13 and 4.15 that we present below.
For every α < κ + let x α := x(α, 1, 1), . . ., x(α, 1, m 1 ) , . . ., x(α, n, 1) . . ., x(α, n, m n ) and We will prove that {U α : α < κ + } is a weak separation for the subspace {x α : Finally, observe that if m := m 1 + • • • + m n , then the spaces X m1 × • • • × X mn and X m are homeomorphic and, therefore, we deduce that X m admits a weakly separated subspace of size κ + , a contradiction to the relation R * (X) Proof.For the inequality R(X) ≤ c F [X] we will show that if Y is a weakly separated subspace of X, then F [X] admits a cellular family of cardinality |Y |.Let {V y : y ∈ Y } be a weak separation for Y .For each y ∈ Y let us take U y ∈ τ X with V y = U y ∩ Y , and consider the collection {[{y}, U y ] : y ∈ Y }.If y, z ∈ Y are distinct and we assume that [{y}, U y ] ∩ [{z}, U z ] = ∅, then {y, z} ⊆ U y ∩ U z .In this way, {y, z} ⊆ V y ∩ V z and, therefore, y ∈ V z and z ∈ V y ; a contradiction to the weak separation hypothesis.Consequently, The relation c F [X] ≤ R * (X) is evident from Theorem 4.12.On the other hand, since for each n The last inequality is a consequence of the following: a remarkable result by Hajnal and Juhász states that |X| ≤ 2 s(X)•ψ(X) whenever X is T 1 (see [5,Theorem 4.7,p. 20]).Thus, as for Hausdorff spaces s ≤ hd and ψ ≤ hL, it is immediate that Corollary 4.14.Every weakly separated space X satisfies the relations c Proof.Let µ := sup{c F [X n ] : n ∈ N}.First, since for every n ∈ N it is satisfied that R(X n ) ≤ c F [X n ] Theorem 4.13), then R * (X) ≤ µ.Now suppose for an absurdity that µ is strictly greater than κ := R * (X).Use the inequality µ ≥ κ + to find n ∈ N with c F [X n ] ≥ κ + , and let {[F α , U α ] : α < κ + } be a cellular family in F [X n ].Given that the function κ + → N determined by α → |F α | has a fiber of cardinality κ + , we will assume without loss of generality that there is m ∈ N with |F α | = m for each α < κ + .Let {x(α, 1), . . ., x(α, m)} be an enumeration without repetitions of F α , and for each 1 For every α < κ + let x α := x(α, 1, 1), . . ., x(α, 1, n) , . . ., x(α, m, 1) . . ., x(α, m, n) and We will see that {U α : α < κ + } is a weak separation for the subspace {x α : Lastly notice that, since the spaces X n × • • • × X n and X mn are homeomorphic, X mn contains a weakly separated subspace of cardinality κ + , a contradiction to the inequality R * (X) < κ + .2) is that all finite powers of a weakly separated space also possess the same characteristic.Now, a natural question that might arise is when the hyperspace F [X] is weakly separated.It turns out that this property is always present.Indeed, simple reasoning shows that {[F, X] : then the following result is verified.Proposition 4.17.If X is a topological space, then for each n ∈ N it is satisfied that F n [X], F [X n ] and F [X] n are weakly separated.
In particular, Proposition 4.17 allows us to detect by means of the weak separation property when a space X does not embed topologically into one of the spaces In particular, by Corollary 4.2 any space X with nw(X) < |X| satisfies the hypothesis of Corollary 4.18.Thus, for example, R does not embed into On the other hand, although there are many difficulties in doing the explicit computation of c F [X] for an arbitrary topological space X, Corollary 4.14 and Proposition 4.17 allow us to do the corresponding calculation for each F n [X] with n ≥ 2.
Corollary 4.19.If X is a topological space and n ∈ N, then Back to condition C(κ), Theorem 4.12 also allows us to refine the first inequality of (3.2) in a natural way.Corollary 4.20.If κ is a cardinal number and X is a topological space with ω < cf(κ) < R * (X), then X does not satisfy C(κ).
Proof.The relations cf(κ) < R * (X) = c * F [X] (see Theorem 4.12) produce a cellular family in F [X] of cardinality cf(κ).For this reason, Proposition 2.2 and 2.3 imply that κ is not a weak precaliber for F [X]. Thus, Theorem 3.11 ensures that X does not satisfy C(κ).
Therefore, the second inequality of (3.2) and Corollary 4.20 imply that, to determine if X satisfies C(κ) or not, we need to focus on those cardinal numbers κ with cf(κ) > ω that satisfy the relations Question 4.21.Is it true that if κ is a regular cardinal and X is a T 1 -space with R * (X) ≤ κ ≤ nw(X), then X satisfies C(κ)?
At this point the answer to Question 4.21 depends on the topological space.For example, the ordinal space [0, ω 1 ) is right-separated with the natural order; in particular, it is weakly separated.For this reason, Corollary 4.14 implies the equalities ] admits a cellular family of cardinality ω 1 and, therefore, Proposition 2.2 guarantees that F [[0, ω 1 )] does not have weak precaliber ω 1 ; consequently, Theorem 3.11 certifies that [0, ω 1 ) does not satisfy C(ω 1 ).

The C(κ) condition in topological products
One question that can be found tacitly in the literature is whether given a pair of spaces X and Y it is satisfied that [2] and [25]).We will give a way to detect when, for a family of topological spaces {X α : α < λ} that fulfills certain properties, the space α<λ F [X α ] is not homeomorphic to F α<λ X α via its chain conditions (see Theorems 5.7, 5.8 and 5.9).
Regarding the relationship between C(κ) and topological products, we start with the following result which follows directly from Lemma 3.7 (recall that the natural projections are continuous and surjective).Proposition 5.1.Let κ and λ be a pair of cardinal numbers (λ not necessarily infinite).If {X α : α < λ} is a family of topological spaces such that the topological product {X α : α < λ} satisfies C(κ), then each factor also satisfies it.
Furthermore, the condition C(κ) is also preserved under finite products as we will see next.
Proposition 5.2.Let κ be a cardinal number.If X and Y are topological spaces that satisfy C(κ), then the product X × Y satisfies it too.
Proof.Let {W α : α < κ} be a subset of τ + X×Y and for each α < κ take In this way, an inductive argument can be used to prove the following corollary.With respect to infinite products, it turns out that it is possible to produce examples to verify that in general the condition C(κ) is not preserved.Recall that t denotes the cardinal function known as tightness (see [5]).

It remains to analyze
what is the answer to the question: will it be possible to determine by means of calibers, precalibers and weak precalibers if given a family of spaces {X α : α < λ}, then the spaces α<λ F [X α ] and F α<λ X α are not homeomorphic?We show below that the answer to the previous question is negative for spaces of the form Proposition 5.5.If X and Y are topological spaces, then the following equalities are true: Proof.For the first equality we note that Theorem 3.3 implies the relations Now, since the argument for precalibers and weak precalibers is similar, we will only expose the details for precalibers.Naturally, to do this we only have to restrict ourselves to elements of UC.On the one hand, if κ ∈ P F [X × Y ] then Theorem 3.11 implies that X × Y satisfies C(κ) and hence Proposition 5.1 ensures that X and Y satisfy C(κ).Thus, Theorem 3.11 guarantees that F [X] and F [Y ] have precaliber κ.On the other hand, if κ ∈ P F [X] ∩ P F [Y ] , Theorem 3.11 says that X and Y satisfy C(κ).Thus, Proposition 5.2 certifies that X × Y satisfies C(κ) and therefore Theorem 3.11 asserts that κ is a precaliber for F [X × Y ].
Clearly Proposition 5.5 can be generalized to any finite product in the natural way.Now, it is necessary to remember that there are several results in the literature that talk about the preservation of precalibers in topological products.For example, the following theorem compiles some of the work done in [14], [19] and [21].
Theorem 5.6.Let κ be an infinite cardinal, λ a cardinal number, {X α : α < λ} a family of topological spaces and X a topological space.
With this result at hand we are ready to prove the following set of theorems.
The following theorem can be proven analogously to the previous one.
Theorem 5.8.Let κ and λ be a pair of cardinal numbers with ω < cf(κ) < λ.If X is a Hausdorff space with more than one point such that nw(X) < cf(κ), then Theorem 5.9.If X is a T 1 -space and λ is a cardinal number (not necessarily infinite), then Furthermore, when |X| λ > |X| the previous inclusion is proper.
Proof.Let us first note that by Corollary 2.4 and Theorem 5.
In particular, Theorems 5.7, 5.8, and 5.9 show that chain conditions can also be used to detect when α<λ F [X α ] and F α<λ X α are not homeomorphic.
6.The weak Lindelöf degree of F If X is a topological space, then the weak Lindelöf degree of X, wL(X), is the cardinal number min We shall say that X is weakly Lindelöf if wL(X) = ω.Additionally, we will work with the cardinal functions In [17,Question 3.23,p. 3087] Sakai asked whether the fact that F [X] is weakly Lindelöf implies that X is hereditarily separable and proved that if X is countably tight then the previous question has an affirmative answer.We shall expand Sakai's result by proving that if F [X] is weakly Lindelöf and X satisfies any of the following conditions: • X is a Hausdorff k-space; • X is a countably tight T 1 -space; • X is weakly separated, then X is hereditarily separable (see Corollary 6.16).What follows is intended to show that if X is a countably tight T 1 -space or X is a Hausdorff k-space, then We will prove (6.1) by establishing generalizations and connections between various results of Sakai and Tall exposed in [17] and [22] respectively.
If C is a collection of subsets of X, we say that C is an ω-cover for X if for any F ∈ [X] <ω there exists C ∈ C with F ⊆ C. In [4] Gerlits and Nagy showed that if X is a topological space, then X n is Lindelöf for every n ∈ N if and only if all ω-open covers of X admit a countable ω-subcover.The following lemma generalizes this fact and its proof can be found in [24, S. 148, p. 122].Lemma 6.1.If X is a topological space and κ is an infinite cardinal, then L * (X) ≤ κ if and only if every open ω-cover of X admits an ω-subcover of size at most κ.
A family A formed by subsets of a space X is discrete if for any x ∈ X there exists U ∈ τ X with x ∈ U and |{A ∈ A : A ∩ U = ∅}| ≤ 1 (see [3, p. 16]).The discrete cellularity of X will be the cardinal number dc(X) := sup |U | : U ⊆ τ + X is a discrete family + ω.
According to [17, Definition 3.1, p. 3083], we say that X satisfies the discrete countable chain condition (in symbols, dccc) if dc(X) = ω.An immediate observation is that if X is a topological space and A is a clopen subset of X, then dc(A) ≤ dc(X).Also, any discrete family U ⊆ τ + X is a cellular family; consequently, it is satisfied that dc(X) ≤ c(X).Furthermore, simple reasoning shows that wL(X) ≤ c(X) (see [5, p. 16]).In the realm of T 3 -spaces we can also connect dc(X) with wL(X).
Proof.Set κ := wL(X).Let λ be an infinite cardinal, U := {U α : α < λ} ⊆ τ + X a discrete family, and for each α < λ let Observe that, since {U α : α < λ} is a discrete family, then {V α : α < λ} is also a discrete family; in particular, since {V α : α < λ} is locally finite it satisfies that V := X \ α<λ V α is an open subset of X (see [3, Corollary 1.1.12,p. 17]).Thus, the collection U ∪ {V } is an open cover of X and thus there exists W ∈ [U ∪ {V }] ≤κ with W = X.Our goal now is to check that U is a subset of W .It turns out that if α < λ then there exists W ∈ W with V α ∩ W = ∅.Then, since V α and V are disjoint sets, necessarily W ∈ U .Finally, since V α ⊆ U α and U is a cellular family, it is verified that U α = W and, therefore, that U α ∈ W .In conclusion, |U | ≤ κ.
The following pair of lemmas can be proven with routine arguments (the second one can be found in [17, Lemma 3.9, p. 3084]).
Proof.Set κ := dc F [X] .To see that L * (X) ≤ κ, let {U α : α < λ} be an open ω-cover of the space X.For each α < λ consider the set then there exists α ∈ J with F ∈ V α and therefore F ⊆ U α .By virtue of Lemma 6.1, this argument proves that L * (X) ≤ κ.
To verify that hd(X) ≤ κ we observe that if U is an open subset of X, then the identity function is a homeomorphism between F [U ] and V ({U }).Thus, since V ({U }) is an open and closed subset of F [X] it satisfies that dc F [U ] ≤ κ and so, the first part of this result guarantees that L * (U ) ≤ κ; in particular, L(U ) ≤ κ.In conclusion, Lemma 6.3 ensures that hL(X) ≤ κ.
Since F [X] is a zero-dimensional Hausdorff space when X is T 1 , Lemma 6.2 and Theorem 6.5 produce the following result.
We now trace the path to prove that hd(X) ≤ wL F [X] if X is a Hausdorff k-space.What follows is to expose some generalizations of the results of the third section of [22].Proposition 6.7.If X is a topological space and d(X) = |X|, or t(X) < cf d(X) , then cf d(X) and d(X) are not calibers for X.
Recall that a topological space X is a k-space if for any A ⊆ X it is satisfied that A is closed in X provided that for any compact subset K of X, A ∩ K is closed in K.In [20] Šapirovskiȋ proved that if X is a Hausdorff k-space, then t(X) ≤ s(X).Our next corollary follows from this result.Corollary 6.10.If X is a Hausdorff k-space, κ is a cardinal number, and all closed subspaces of X have caliber κ + , then hd(X) ≤ κ.
Proof.Under our hypotheses it is satisfied that all the closed subspaces of X have cellularity less than κ + ; consequently, all its subspaces have cellularity less than κ + and, therefore, s(X) ≤ κ.Finally, since t(X) ≤ κ, Theorem 6.9 guarantees that hd(X) ≤ κ.Theorem 6.12.If X is a Hausdorff k-space or X has countable tightness, then hd(X) ≤ wL F [X] .
Proof.Set κ := wL F [X] .If X has countable tightness, then from (6.2) it follows that hd(X) = hd c (X).Thus, Lemma 6.11 ensures that hd(X) ≤ κ.Now, if X is a Hausdorff k-space, then Lemma 6.11 implies that any closed subspace of X has density at most κ; consequently, Proposition 2.1 guarantees that all closed subspaces of X have caliber κ + and hence, Corollary 6.10 certifies that hd(X) ≤ κ.
In conclusion, Theorems 6.6 and 6.12 produce the following corollaries.Corollary 6.13.If X is a Hausdorff k-space or X is a T 1 -space with countable tightness, then The connections between some of the cardinal functions of X and F Naturally, in the relation wL F [X] → hd(X) we need X to be T 1 and have countable tightness, or else to be a Hausdorff k-space.
In general, we don't know if it is possible to directly connect R(X) and dc F [X] in some way, but we can establish a relationship in this regard by considering a variant for the R function.The following result will be essential to achieve this objective.

Example 4 . 11 .
If R is the space obtained by equipping the set R with the topology generated by the subbase S := {U \ S : U ∈ τ R and S is a non-trivial convergent sequence in R},

Question 4 . 16 .
Are there examples of spaces X and Y such that R(X) < c F [X] and c F [Y ] < R * (Y )?A consequence of Proposition 4.8(

Corollary 5 . 3 .
Let κ be a cardinal number.If 1 ≤ n < ω, {X m : m < n} is a family of topological spaces and each X m satisfies C(κ), then {X m : m < n} satisfies it too.

A
cardinal function φ reflects a cardinal number κ if the condition φ(X) ≥ κ implies the existence of Y ∈ [X] ≤κ with φ(Y ) ≥ κ.For example, a classical result states that density reflects any regular cardinal, that is, if κ is a regular cardinal and d(X) ≥ κ, then there exists Y ⊆ X with |Y | = d(Y ) = κ (see [6, Theorem 2.5, p. 54]).The next lemma follows from this fact.Lemma 6.8.If X is a topological space such that hd(X) > κ, then there exists Y ⊆ X with |Y | = d(Y ) = κ + .

Figure 1 .
Figure 1.Relations between some cardinal functions of X and F [X]