A common extension of Arhangel'skii's Theorem and the Hajnal-Juhasz inequality

TL;DR

This paper introduces a unified bound for the weak Lindelöf number of the $G_\delta$-modification of Hausdorff spaces, generalizing key cardinal inequalities in topology without requiring separation axioms beyond $T_2$.

Contribution

It provides a common extension of Arhangel'skii's Theorem and the Hajnal-Juhasz inequality, answering a longstanding open question in cardinal invariants.

Findings

Unified bound for the weak Lindelöf number of $G_\delta$-modification

Generalizes Arhangel'skii's and Hajnal-Juhasz inequalities

Does not require separation axioms beyond $T_2$

Abstract

We present a bound for the weak Lindel\"of number of the $G_\delta$-modification of a Hausdorff space which implies various known cardinal inequalities, including the following two fundamental results in the theory of cardinal invariants in topology: $|X|\le 2^{L(X)\chi(X)}$ (Arhangel'skii) and $|X|\le 2^{c(X)\chi (X)}$ (Hajnal-Juhasz). This solves a question that goes back to Bell, Ginsburg and Woods and is mentioned in Hodel's survey on Arhangel'skii's Theorem. In contrast to previous attempts we do not need any separation axiom beyond $T_2$.

Equations

$$M\models X\subset \bigcup \{\overline{\bigcup {\mathcal{V}}_{\alpha }}:\alpha <\kappa \}$$

$$H(\theta )\models X\subset \bigcup \{\overline{\bigcup {\mathcal{V}}_{\alpha }}:\alpha <\kappa \}$$

Full text

The research that led to the present paper was partially supported by a grant of the group GNSAGA of INdAM

1. Introduction

Two of the milestones in the theory of cardinal invariants in topology are the following inequalities:

Theorem 1**.**

(Arhangel’skiĭ, 1969) [2] If $X$ is a $T_{2}$ space, then $|X|\leq 2^{L(X)\chi(X)}$.

Theorem 2**.**

(Hajnal-Juhász, 1967) [13] If $X$ is a $T_{2}$ space, then $|X|\leq 2^{c(X)\chi(X)}$.

Here $\chi(X)$ denotes the character of $X$, $c(X)$ denotes the cellularity of $X$, that is the supremum of the cardinalities of the pairwise disjoint collection of non-empty open subsets of $X$ and $L(X)$ denotes the Lindelöf degree of $X$, that is the smallest cardinal $\kappa$ such that every open cover of $X$ has a subcover of size at most $\kappa$.

The intrinsic difference between the cellularity and the Lindelöf degree makes it non-trivial to find a common extension of the two previous inequalities. The first attempt was done in 1978 by Bell, Ginsburg and Woods [6], who used the notion of weak Lindelöf degree. The weak Lindelöf degree of $X$ ($wL(X)$) is defined as the least cardinal $\kappa$ such that every open cover of $X$ has a $\leq\kappa$-sized subcollection whose union is dense in $X$. Clearly, $wL(X)\leq L(X)$ and we also have $wL(X)\leq c(X)$, since every open cover without $<\kappa$-sized dense subcollections can be refined to a $\kappa$-sized pairwise disjoint family of non-empty open sets by an easy transfinite induction. Unfortunately, the Bell-Ginsburg-Woods result needs a separation axiom which is much stronger than Hausdorff.

Theorem 3**.**

[6]** If $X$ is a normal space, then $|X|\leq 2^{wL(X)\chi(X)}$.

It is still unknown whether this inequality is true for regular spaces, but in [6] it was shown that it may fail for Hausdorff spaces. Indeed, the authors constructed Hausdorff non-regular first-countable weakly Lindelöf spaces of arbitrarily large cardinality.

Arhangel’skiĭ [3] got closer to obtaining a common generalization of these two fundamental results by introducing a relative version of the weak Lindelöf degree, namely the cardinal invariant $wL_{c}(X)$, i.e. the least cardinal $\kappa$ such that for any closed set $F$ and any family of open sets $\mathcal{U}$ satisfying $F\subseteq\bigcup\mathcal{U}$ there is a subcollection $\mathcal{V}\in[\mathcal{U}]^{\leq\kappa}$ such that $F\subseteq\overline{\bigcup\mathcal{V}}$.

Theorem 4**.**

[3]** If $X$ is a regular space, then $|X|\leq 2^{wL_{c}(X)\chi(X)}$.

O. Alas [1] showed that the previous inequality continues to hold for Urysohn spaces, but it is still open whether it’s true for Hausdorff spaces.

In [4] Arhangel’skii made another step ahead by introducing the notion of strict quasi-Lindelöf degree, which allowed him to give a common refinement of the countable case of his 1969 theorem and the Hajnal-Juhász inequality. He defined a space $X$ to be strict quasi-Lindelöf if for every closed subset $F$ of $X$, for every open cover $\mathcal{U}$ of $F$ and for every countable decomposition $\{\mathcal{U}_{n}:n<\omega\}$ of $\mathcal{U}$ there are countable subfamilies $\mathcal{V}_{n}\subset\mathcal{U}_{n}$, for every $n<\omega$ such that $F\subset\bigcup\{\overline{\bigcup\mathcal{V}_{n}}:n<\omega\}$. It is easy to see that every Lindelöf space is strict quasi-Lindelöf and every ccc space is strict-quasi Lindelöf. Arhangel’skii proved that every strict quasi-Lindelöf first-countable space has cardinality at most continuum.

However, Arhangel’skii’s approach cannot be extended to higher cardinals. Indeed, it’s not even clear whether $|X|\leq 2^{\chi(X)}$ is true for every strict quasi-Lindelöf space $X$. This inspired us to introduce the following cardinal invariants:

Definition 5**.**

• •

The piecewise weak Lindelöf degree of $X$ ($pwL(X)$) is defined as the minimum cardinal $\kappa$ such that for every open cover $\mathcal{U}$ of $X$ and every decomposition $\{\mathcal{U}_{i}:i\in I\}$ of $\mathcal{U}$, there are $\leq\kappa$-sized families $\mathcal{V}_{i}\subset\mathcal{U}_{i}$, for every $i\in I$ such that $X\subset\bigcup\{\overline{\bigcup\mathcal{V}_{i}}:i\in I\}$.

• •

The piecewise weak Lindelöf degree for closed sets of $X$ ($pwL_{c}(X)$) is defined as the minimum cardinal $\kappa$ such that for every closed set $F\subset X$, for every open family $\mathcal{U}$ covering $F$ and for every decomposition $\{\mathcal{U}_{i}:i\in I\}$ of $\mathcal{U}$, there are $\leq\kappa$-sized subfamilies $\mathcal{V}_{i}\subset\mathcal{U}_{i}$ such that $F\subset\bigcup\{\overline{\bigcup\mathcal{V}_{i}}:i\in I\}$.

As a corollary to our main result, we will obtain the following bound, which is the desired common extension of Arhangel’skii’s Theorem and the Hajnal-Juhász inequality.

Theorem 6**.**

For every Hausdorff space $X$, $|X|\leq 2^{pwL_{c}(X)\cdot\chi(X)}$.

For undefined notions we refer to [11]. Our notation regarding cardinal functions mostly follows [14]. To state our proofs in the most elegant and compact way we use the language of elementary submodels, which is well presented in [10].

2. A cardinal bound for the $G_{\delta}$-modification

The following proposition collects a few simple general facts about the piecewise weak Lindelöf number which will be helpful in the proof of the main theorem.

Proposition 7**.**

For any space $X$ we have:

1. (1)

$pwL(X)\leq pwL_{c}(X)$. 2. (2)

$pwL_{c}(X)\leq L(X)$. 3. (3)

$pwL_{c}(X)\leq c(X)$. 4. (4)

If $X$ is $T_{3}$ then $wL_{c}(X)\leq pwL(X)$.

Proof.

The first two items are trivial. To prove the third item, let $F$ be a closed subset of $X$ and $\mathcal{V}=\bigcup\{\mathcal{V}_{i}:i\in I\}$ be an open collection satisfying $F\subseteq\bigcup\mathcal{V}$. Suppose $c(X)\leq\kappa$. For every $i\in I$ let $\mathcal{C}_{i}$ be a maximal collection of pairwise disjoint non-empty open subsets of $X$ such that for each $C\in\mathcal{C}_{i}$ there is some $V_{C}\in\mathcal{V}_{i}$ with $C\subseteq V_{C}$. By letting $\mathcal{W}_{i}=\{V_{C}:C\in\mathcal{C}_{i}\}$, the maximality of $\mathcal{C}_{i}$ implies that $\bigcup\mathcal{V}_{i}\subseteq\overline{\bigcup\mathcal{W}_{i}}$ and so $F\subseteq\bigcup\{\overline{\cup\mathcal{W}_{i}}:i\in I\}$. Since $|\mathcal{W}_{i}|\leq|\mathcal{C}_{i}|\leq\kappa$, we have $pwL_{c}(X)\leq\kappa$.

To prove the fourth item assume $X$ is a regular space and let $\kappa$ be a cardinal such that $pwL(X)\leq\kappa$. Let $F$ be a closed subset of $X$ and $\mathcal{U}$ be an open cover of $F$. If $\mathcal{U}$ covers $X$ we’re done. Otherwise use regularity to choose, for every $p\in X\setminus\bigcup\mathcal{U}$ an open set $U_{p}$ such that $p\in U_{p}$ and $F\cap\overline{U}_{p}=\emptyset$. Note that $\mathcal{U}\cup\{U_{p}:p\in X\setminus F\}$ is an open cover of $X$, so by $pwL(X)\leq\kappa$, there is a $\kappa$-sized subfamily $\mathcal{V}$ of $\mathcal{U}$ such that $X\subset\overline{\bigcup\mathcal{V}}\cup\bigcup\{\overline{U_{p}}:p\in X\setminus F\}$. Hence $F\subset\overline{\bigcup\mathcal{V}}$ and we are done. ∎

Corollary 8**.**

If $X$ is a regular space then $|X|\leq 2^{pwL(X)\cdot\chi(X)}$.

Proof.

Combine Proposition 7, (4) and Arhangel’skii’s result that $|X|\leq 2^{wL_{c}(X)\cdot\chi(X)}$ for every regular space $X$. ∎

We state our main theorem in terms of the $G_{\kappa}$-modification of a space. Let $\kappa$ be a cardinal number. By $X_{\kappa}$ we denote the topology on $X$ generated by $\kappa$-sized intersections of open sets of $X$. We call $X_{\kappa}$, the $G_{\kappa}$-modification of $X$; in case $\kappa=\omega$ we speak of the $G_{\delta}$-modification of $X$ and we often use the symbol $X_{\delta}$ instead. This construction has been extensively studied in the literature; various authors have tried to bound the cardinal functions of $X_{\kappa}$ in terms of their values on $X$ (see, for example [8], [12], [16], [17], [18]) and results of this kind have found applications to other topics in topology, like the estimation of the cardinality of compact homogeneous spaces (see [5], [8], [9] and [18]).

By $X^{c}_{\kappa}$ we denote the topology on $X$ generated by $G^{c}_{\kappa}$-sets, that is those subsets $G$ of $X$ such that there is a family $\{U_{\alpha}:\alpha<\kappa\}$ of open sets with $G=\bigcap\{U_{\alpha}:\alpha<\kappa\}=\bigcap\{\overline{U_{\alpha}}:\alpha<\kappa\}$. In general, the topology of $X^{c}_{\kappa}$ is coarser than the $G_{\kappa}$-modification of $X$, but if $X$ is a regular space then $X^{c}_{\kappa}=X_{\kappa}$.

Theorem 9**.**

Let $X$ be a Hausdorff space such that $t(X)\cdot pwL_{c}(X)\leq\kappa$ and $X$ has a dense set of points of character $\leq\kappa$. Then $wL(X^{c}_{\kappa})\leq 2^{\kappa}$.

Proof.

Let $\mathcal{F}$ be a cover of $X$ by $G^{c}_{\kappa}$-sets. Let $\theta$ be a large enough regular cardinal and $M$ be a $\kappa$-closed elementary submodel of $H(\theta)$ such that $|M|=2^{\kappa}$ and $M$ contains everything we need (that is, $X,\mathcal{F}\in M$, $\kappa+1\subset M$ etc…).

For every $F\in\mathcal{F}$ choose open sets $\{U_{\alpha}:\alpha<\kappa\}$ such that $F=\bigcap\{U_{\alpha}:\alpha<\kappa\}=\bigcap\{\overline{U_{\alpha}}:\alpha<\kappa\}$.

Claim 1. $\mathcal{F}\cap M$ covers $\overline{X\cap M}$.

Proof of Claim 1.

Let $x\in\overline{X\cap M}$. Since $\mathcal{F}$ is a cover of $X$ we can find a set $F\in\mathcal{F}$ such that $x\in F$. Moreover, using $t(X)\leq\kappa$, we can find a $\kappa$-sized subset $S$ of $X\cap M$ such that $x\in\overline{S}$. Note that $x\in\overline{U_{\alpha}\cap S}$, for every $\alpha<\kappa$. Moreover, by $\kappa$-closedness of $M$, the set $U_{\alpha}\cap S$ belongs to $M$. Set $B=\bigcap\{\overline{U_{\alpha}\cap S}:\alpha<\kappa\}$. Note that $x\in B\subset F$ and $B\in M$. Therefore $H(\theta)\models(\exists G\in\mathcal{F})(x\in B\subset G)$ and all the free variables in the previous formula belong to $M$. Therefore, by elementarity we also have that $M\models(\exists G\in\mathcal{F})(x\in B\subset G)$ and hence there exists a set $G\in\mathcal{F}\cap M$ such that $x\in G$, which is what we wanted to prove.

∎

Claim 2. $\mathcal{F}\cap M$ has dense union in $X$.

Proof of Claim 2.

Suppose by contradiction that $X\nsubseteq\overline{\bigcup(\mathcal{F}\cap M)}$. Then we can fix a point $p\in X\setminus\overline{\bigcup(\mathcal{F}\cap M)}$ such that $\chi(p,X)\leq\kappa$. Let $\{V_{\alpha}:\alpha<\kappa\}$ be a local base at $p$.

For every $F\in\mathcal{F}\cap M$, let $\{U_{\alpha}(F):\alpha<\kappa\}\in M$ be a sequence of open sets such that $F=\bigcap\{U_{\alpha}(F):\alpha<\kappa\}=\bigcap\{\overline{U_{\alpha}(F)}:\alpha<\kappa\}$. Note that $\{U_{\alpha}(F):\alpha<\kappa\}\subset M$. Let $\mathcal{C}=\{U_{\alpha}(F):F\in\mathcal{F}\cap M,\alpha<\kappa\}$. Note that $\mathcal{C}$ is an open cover of $\overline{X\cap M}$ and $\mathcal{C}\subset M$.

For every $x\in\overline{X\cap M}$, we can choose, using Claim 1, a set $F_{x}\in\mathcal{F}\cap M$ such that $x\in F_{x}$. Since $p\notin F_{x}$, there is $\alpha<\kappa$ such that $p\notin\overline{U_{\alpha}(F_{x})}$. Hence we can find an ordinal $\beta_{x}<\kappa$ such that $V_{\beta_{x}}\cap U_{\alpha}(F_{x})=\emptyset$. This shows that $\mathcal{U}=\{U\in\mathcal{C}:(\exists\beta<\kappa)(U\cap V_{\beta}=\emptyset)\}$ is an open cover of $\overline{X\cap M}$. Let $\mathcal{U}_{\alpha}=\{U\in\mathcal{U}:U\cap V_{\alpha}=\emptyset\}$. Then $\{\mathcal{U}_{\alpha}:\alpha<\kappa\}$ is a decomposition of $\mathcal{U}$ and hence we can find a $\kappa$-sized family $\mathcal{V}_{\alpha}\subset\mathcal{U}_{\alpha}$ for every $\alpha<\kappa$ such that $\overline{X\cap M}\subset\bigcup\{\overline{\bigcup\mathcal{V}_{\alpha}}:\alpha<\kappa\}$. Note that by $\kappa$-closedness of $M$ the sequence $\{\overline{\bigcup\mathcal{V}_{\alpha}}:\alpha<\kappa\}$ belongs to $M$ and hence the previous formula implies that:

[TABLE]

So, by elementarity:

[TABLE]

But that is a contradiction, because $p\notin\overline{\bigcup\mathcal{V}_{\alpha}}$, for every $\alpha<\kappa$.

∎

Since $|\mathcal{F}\cap M|\leq 2^{\kappa}$, Claim 2 proves that $wL(X^{c}_{\kappa})\leq 2^{\kappa}$, as we wanted.

∎

As a first consequence, we derive the desired common extension of Arhangel’skii’s Theorem and the Hajnal-Juhász inequality.

Recall that the closed pseudocharacter of the point $x$ in $X$ ($\psi_{c}(x,X)$) is defined as the minimum cardinal $\kappa$ such that there is a $\kappa$-sized family $\{U_{\alpha}:\alpha<\kappa\}$ of open neighbourhoods of $x$ with $\bigcap\{\overline{U_{\alpha}}:\alpha<\kappa\}=\{x\}$. The closed pseudocharacter of $X$ ($\psi_{c}(X)$) is then defined as $\psi_{c}(X)=\sup\{\psi_{c}(x,X):x\in X\}$.

Corollary 10**.**

Let $X$ be a Hausdorff space. Then $|X|\leq 2^{pwL_{c}(X)\cdot\chi(X)}$.

Proof.

It suffices to note that in a Hausdorff space $\psi_{c}(X)\cdot t(X)\leq\chi(X)$ and hence if $\kappa$ is a cardinal such that $\chi(X)\leq\kappa$ then $X^{c}_{\kappa}$ is a discrete set. Thus $wL(X^{c}_{\kappa})\leq 2^{\kappa}$ if and only if $|X|=|X^{c}_{\kappa}|\leq 2^{\kappa}$. ∎

Remark. Corollary 10 is a strict improvement of both Arhangel’skii’s Theorem and the Hajnal-Juhász inequality. Indeed, if $S$ is the Sorgenfrey line and $A([0,1])$ the Aleksandroff duplicate of the unit interval, then the space $X=(S\times S)\oplus A([0,1])$ is first countable, $pwL_{c}(X)=\aleph_{0}$ and $L(X)=c(X)=\mathfrak{c}$.

Recall that a space is initially $\kappa$-compact if every open cover of cardinality $\leq\kappa$ has a finite subcover (for $\kappa=\omega$ we obtain the usual notion of countable compactness). The following Lemma essentially says that if $X$ is an initially $\kappa$-compact spaces such that $wL_{c}(X)\leq\kappa$, then it satisfies the definition of $pwL_{c}(X)\leq\kappa$ when restricted to decompositions of cardinality at most $\kappa$.

Lemma 11**.**

Let $X$ be an initially $\kappa$-compact space such that $wL_{c}(X)\leq\kappa$ and $F$ be a closed subset of $X$. If $\mathcal{U}$ is an open cover of $F$ and $\{\mathcal{U}_{\alpha}:\alpha<\kappa\}$ is a $\kappa$-sized decomposition of $\mathcal{U}$, then there are $\kappa$-sized subfamilies $\mathcal{V}_{\alpha}\subset\mathcal{U}_{\alpha}$ such that $F\subset\bigcup\{\overline{\bigcup\mathcal{V}_{\alpha}}:\alpha<\kappa\}$

Proof.

Let $U_{\alpha}=\bigcup\mathcal{U}_{\alpha}$. Then $\{U_{\alpha}:\alpha<\kappa\}$ is an open cover of $F$ of cardinality $\kappa$, so by initial $\kappa$-compactness there is a finite subset $S$ of $\kappa$ such that $F\subset\{U_{\alpha}:\alpha\in S\}$. Let now $\mathcal{W}=\bigcup\{\mathcal{U}_{\alpha}:\alpha\in S\}$. We then have $F\subset\bigcup\mathcal{W}$ and hence by $wL_{c}(X)\leq\kappa$ we can find a $\kappa$-sized subfamily $\mathcal{W}^{\prime}$ of $\mathcal{W}$ such that $F\subset\overline{\bigcup\mathcal{W}^{\prime}}$. Set now $\mathcal{V}_{\alpha}=\{W\in\mathcal{W}^{\prime}:W\in\mathcal{U}_{\alpha}\}$. Then $|\mathcal{V}_{\alpha}|\leq\kappa$ and $F\subset\bigcup\{\overline{\bigcup\mathcal{V}_{\alpha}}:\alpha<\kappa\}$, as we wanted. ∎

Noticing that in the proof of Theorem 9 we only needed to apply the definition of $pwL_{c}(X)\leq\kappa$ to decompositions of cardinality $\kappa$, Theorem 9 and Lemma 11 imply the following corollaries.

Corollary 12**.**

[8]** Let $X$ be an initially $\kappa$-compact space containing a dense set of points of character $\leq\kappa$ and such that $wL_{c}(X)\cdot t(X)\leq\kappa$. Then $wL(X^{c}_{\kappa})\leq 2^{\kappa}$.

Corollary 13**.**

(Alas, [1]) Let $X$ be an initially $\kappa$-compact space with a dense set of points of character $\kappa$, such that $wL_{c}(X)\cdot t(X)\cdot\psi_{c}(X)\leq\kappa$. Then $|X|\leq 2^{\kappa}$.

3. Open Questions

Corollary 8 can be slightly improved by replacing regularity with the Urysohn separation property (that is, every pair of distinct points can be separated by disjoint closed neighbourhoods). Indeed, in a similar way as in the proof of Proposition 7 (4) it can be shown that if $X$ is Urysohn then $wL_{\theta}(X)\leq pwL(X)$, where $wL_{\theta}(X)$ is the weak Lindelöf number for $\theta$-closed sets (see [7]). Moreover, $|X|\leq 2^{wL_{\theta}(X)\cdot\chi(X)}$ for every Urysohn space $X$. However it’s not clear whether regularity can be weakened to the Hausdorff separation property. That motivates the next question.

Question 3.1**.**

Is the inequality $|X|\leq 2^{pwL(X)\cdot\chi(X)}$ true for every Hausdorff space $X$?

Moreover, we were not able to find an example which distinguishes countable piecewise weak Lindelöf number for closed sets from the strict quasi-Lindelöf property.

Question 3.2**.**

Is there a strict quasi-Lindelöf space $X$ such that $pwL_{c}(X)>\aleph_{0}$?

Finally, Arhangel’skii’s notion of a strict quasi-Lindelöf space suggests a natural cardinal invariant. Define the strict quasi-Lindelöf number of $X$ ($sqL(X)$) to be the least cardinal number $\kappa$, such that for every closed subset $F$ of $X$, for every open cover $\mathcal{U}$ of $F$ and for every $\kappa$-sized decomposition $\{\mathcal{U}_{\alpha}:\alpha<\kappa\}$ of $\mathcal{U}$ there are $\kappa$-sized subfamilies $\mathcal{V}_{\alpha}\subset\mathcal{U}_{\alpha}$ such that $X\subset\bigcup\{\overline{\bigcup\mathcal{V}_{\alpha}}:\alpha<\kappa\}$. Obviously $sqL(X)\leq pwL_{c}(X)$. It’s not at all clear from our argument whether the piecewise weak-Lindelöf number for closed sets can be replaced with the strict quasi-Lindelöf number in Corollary 10.

Question 3.3**.**

Let $X$ be a Hausdorff space. Is it true that $|X|\leq 2^{sqL(X)\cdot\chi(X)}$?

Even the following special case of the above question seems to be open.

Question 3.4**.**

Let $X$ be a strict quasi-Lindelöf space. Is it true that $|X|\leq 2^{\chi(X)}$?