Special sets of reals and weak forms of normality
on Isbell-Mrówka spaces

Given a countable infinite set $N$, an almost disjoint family (on $N$) is a infinite collection of infinite subsets of $N$ whose pairwise intersections are finite. Throughout this paper we assume that $[N]^{\omega}\cap N=\emptyset$. A MAD family (maximal almost disjoint family) is an almost disjoint family which is not properly contained in any other family. It is well known that there are almost disjoint families of size $\mathfrak{c}$ [1].

Each almost disjoint family $\mathcal{A}$ on $N$ is associated to the Isbell-Mrówka space of $\mathcal{A}$, also called the $Psi$-space of $\mathcal{A}$ and denoted by $\Psi(\mathcal{A})$. This space is the set $N\cup\mathcal{A}$, where $N$ is open and discrete and for each $a\in\mathcal{A}$, the sets of the form $\{a\}\cup(a\setminus F)$, where $F\subseteq N$ is finite, form an open basis for the point $a$. It is easy to verify that $\mathcal{A}$ is a Hausdorff, locally compact (and therefore Tychonoff) zero-dimensional non-compact separable Moore topological space.

The topological properties of $\Psi(\mathcal{A})$ often depend on the combinatorial properties of $\mathcal{A}$. For instance, $\Psi(\mathcal{A})$ is pseudocompact iff $\mathcal{A}$ is MAD and $\Psi(\mathcal{A})$ is metrizable iff $\mathcal{A}$ is countable. $\Psi$-spaces provide a rich source of counter-examples to many topoligical questions including questions about convergence and compactness, and there are many interesting and difficult problems about the combinatorial properties of almost disjoint families that are interesting in their own right. As good introductions to this field of study we mention [10] and [8].

To improve our notation, we say that an almost disjoint family $\mathcal{A}$ has a certain topological property if and only if $\Psi(\mathcal{A})$ has this topological property. So the statement “$\mathcal{A}$ is normal” means the same as “$\Psi(\mathcal{A})$ is normal”.

The normality of Isbell-Mrówka spaces has been extensively studied, perhaps initially in relation to the Normal Moore Space Problem. These problems are closely related to the concept of $Q$-set, a special kind of sets of reals:

We have the following classical result. We give [9] as a reference.

The proof (1)$\leftrightarrow$(2) of the proposition above is sketched as follows: given a set $X\subseteq 2^{\omega}$ of reals, we may define an almost disjoint family on $N=2^{<\omega}$ named $\mathcal{A}_{X}$ consisting of the sets $a_{x}=\{x|_{n}:n\in\omega\}$ for $x\in X$. Then it is shown that $\mathcal{A}_{X}$ is normal iff $X$ is a $Q$-set. Moreover, it is shown that if $\mathcal{A}$ is a normal almost disjoint family on $\omega$, then it is a $Q$-set of the Polish space $\mathcal{P}(\omega)$.

Of course, the existence of a $Q$-set (and so all the above statements) are independent of the axioms of ZFC. For instance, under CH (2) fails since such a space would be a separable normal space with a closed discrete subspace of size $\mathfrak{c}$, which violates Jones’s Lemma, and under $\mathfrak{p}=\mathfrak{c}$, every set of reals of size $<\mathfrak{c}=\mathfrak{p}$ is a $Q$-set (since the natural poset for making a subset of a given set a relative $G_{\delta}$ is $\sigma$-centered, as mentioned in [2]).

The main contribution of this paper is the introduction of a new class of special subsets of the reals which we call weak $\lambda$-set. For information on the classical special subsets of reals, such as $\sigma$-sets, $\lambda$-sets and $Q$-sets we refer to [12].

In the past decades several weakenings of normality have been proposed and studied. A topological space is pseudonormal if for every closed set $K$ and every countable closed set $F$ there exist disjoint open sets $U,V$ such that $F\subseteq U$, $K\subseteq V$. It is mentioned in [9], while a $Q$-set $X\subseteq 2^{\omega}$ gives a normal $\Psi$-space, we have $X\subseteq 2^{\omega}$ is a $\lambda$-set if and only if $\mathcal{A}_{X}$ is pseudonormal. Conversely, if $\mathcal{A}$ is a pseudonormal almost disjoint family, then it is a $\lambda$-set of $\mathcal{P}(\omega)$. The proofs are similar.

Other weakenings of normality have been studied in the realm of Isbell-Mrówka spaces. We cite [7], [4] as recent works.

In [7], several weakenings of normality in Isbell-Mrówka spaces have been studied including the notion of almost-normal. We say that a topological space is almost-normal iff for every closed set $K$ and every regular closed set $F$ disjoint from $K$ there exists two disjoint open sets $U$, $V$ such that $K\subseteq U$ and $F\subseteq V$. They asked if there exists an almost normal Isbell-Mrówka space which is not normal, or, more strongly, if there exists a MAD family whose Isbell-Mrówka space is almost-normal. In [4], V. Rodrigues and V. Ronchim used forcing to show that the answer to the former is consistently positive with both CH and $\neg$CH. The latter is still open.

In [4], in order to produce the mentioned example something which was defined as “almost $Q$-set” was used. This definition, which will be stated in the next section, was designed to work with almost-normality in the same way as $Q$-sets work with normality, that is, in a way such that $X\subseteq 2^{\omega}$ is an almost $Q$-set if and only if $\mathcal{A}_{X}$ is almost-normal. In this paper, we show that this class of sets actually is the well known class of the $\sigma$-sets of reals (see [12]).

In [7], P. Szeptycki and S. Garcia-Balan defined the notion of an strongly $\aleph_{0}$-separated almost disjoint family, which is an almost disjoint family where every pair of disjoint closed countable sets can be separated by a clopen set. They proved that every almost-normal almost disjoint family is strongly $\aleph_{0}$-separated. In this paper we introduce a new class of special subsets of the reals which we call weak $\lambda$-sets which fits between the class of $\lambda$-sets and the class of perfectly meager sets ($X\subseteq 2^{\omega}$ is perfectly meager if and only if its intersection with every perfect set $P$ is meager in $P$). It follows that $X\subseteq 2^{\omega}$ is weak $\lambda$ if and only if $\mathcal{A}_{X}$ is strongly $\aleph_{0}$-separated, and that if an almost disjoint family $\mathcal{A}$ is strongly $\aleph_{0}$-separated then it is a weak $\lambda$-set of $\mathcal{P}(\omega)$ Consistently, there are perfectly meager sets which are not weak $\lambda$-sets. We do not know if there is a weak $\lambda$-set which is not a $\lambda$-set.

As mentioned in the introduction, throughout this paper we will give special attention to the almost disjoint families of branches in the Cantor tree $2^{<\omega}$:

Notice that almost disjoint families of branches are never MAD because each element of $\mathcal{A}_{X}$ intersects each infinite anti-chain in $2^{<\omega}$ at most in one point. It is worth mentioning that for this special class of almost disjoint families some topological properties from $\Psi(\mathcal{A}_{X})$ can be characterized in terms of topological properties the set $X$. The proposition below is probably folklore, but we include the proof for completeness.

Given an almost disjoint family $\mathcal{A}$ over $N$, we say that a set $J\subseteq N$ is a partitioner for disjoint subfamilies $\mathcal{B},\mathcal{C}\subseteq\mathcal{A}$ iff the following are satisfied:

1. 1.

   For all $a\in\mathcal{A}$: $a\subseteq^{*}J$ or $a\cap J=^{*}\emptyset$;

2. 2.

   For all $b\in\mathcal{B}$, $b\subseteq^{*}J$;

3. 3.

   For all $c\in\mathcal{C}$, $c\cap J=^{*}\emptyset$.

It is well known and easy to see that an almost disjoint family $\mathcal{A}$ is normal iff for all $\mathcal{B}\subseteq\mathcal{A}$, there exist a partitioner for $\mathcal{B}$ and $\mathcal{A}\setminus\mathcal{B}$. As an immediate consequence of this fact and the previous result, we obtain the following folklore result:

Recall that an almost disjoint family $\mathcal{A}$ is strongly $\aleph_{0}$-separated iff each pair of countable disjoint subfamilies $\mathcal{B},\mathcal{C}\subseteq\mathcal{A}$ can be separated by a partitioner. This definition was first presented in [7], it is weaker than almost-normality and under CH there exist strongly $\aleph_{0}$-separated MAD families.

In [4, Theorem 3.6] equivalent conditions to almost-normality in $\Psi$-spaces were presented. In particular, the authors have shown that for an arbitrary almost disjoint family $\mathcal{A}$, the following holds:

Combining (2.6) and Proposition 2.5 yields an indirect proof that the almost $Q$-sets defined in [4] are, in fact, the well-known class of $\sigma$-sets:

Recall that a Luzin family (Luzin^∗ family) is an almost-disjoint family $\mathcal{A}$ of size $\omega_{1}$ for which there exists an injective enumeration $\mathcal{A}=\{a_{\alpha}:\alpha<\omega_{1}\}$ such that $\forall\alpha<\omega_{1}\forall n\in\omega\,\{\beta<\alpha:a_{\beta}\cap a_{\alpha}\subseteq n\}$ is finite (such that $\forall\alpha<\omega_{1}\forall n\in\omega\,\{\beta<\alpha:|a_{\beta}\cap a_{\alpha}|<n\}$ is finite). Clearly, every Luzin^∗ family is a Luzin family. Luzin families fail to be normal badly: for every pair of disjoint uncountable sets $\mathcal{B}$, $\mathcal{C}$ of $\mathcal{A}$ there is no $X\subseteq\omega$ such that for every $a\in\mathcal{B}$, $a\subseteq^{*}X$ and $\forall a\in\mathcal{C}$, $a\cap X=^{*}\emptyset$.

In [11], Hrušák and Guzmán introduced the notion of an almost disjoint family potentially having a property $P$. Given a property $P$ of almost disjoint families and an almost disjoint family $\mathcal{A}$, they defined $\mathcal{A}$ is potentially $P$ if there exists a ccc forcing notion $\mathbb{P}$ such that $\mathbbm{1}\Vdash_{\mathbb{P}}\check{\mathcal{A}}\text{ is }P$. They showed that $\mathcal{A}$ is potentially normal iff $\mathcal{A}$ has no $n$-Luzin gap (see their paper for the definition).

We can ask if there is a nice characterization for potentially almost-normal almost disjoint families. We don’t have a answer for this question. However, we have the following:

Such a set $X$ does not exist for every Luzin family. For instance, in Example 2.10 [7], CH is used to construct a MAD Luzin family $\mathcal{A}$ for which for every $X\subseteq\omega$, $\{a\in\mathcal{A}:|a\cap X|=\omega\}$ is either finite or co-countable. It is not clear for us if that Luzin family is potentially almost-normal.

We note that for any Luzin family $\mathcal{A}$ and for any uncountable set $\mathcal{B}\subseteq\mathcal{A}$ whose complement is also uncountable, we can add, by a c.c.c. forcing, a set $X$ such that $\mathcal{B}=\{a\in A:|a\cap X|=\omega\}$, thus:

Notice that $\mathcal{A}$ may potentially have a property $P$ and potentially have property $\neg P$.

In [4] the authors extended the definition of strongly $\aleph_{0}$-separated almost disjoint family, introduced in [7], in the following way: an almost disjoint family $\mathcal{A}$ is said to be strongly $(\aleph_{0},<\!\mathfrak{c})$-separated if and only if every pair of disjoint subfamilies of $\mathcal{A}$ can be separated, provided one is countable and the other has size less than $\mathfrak{c}$. This stronger property is useful to distinguish, at least consistently, almost-normal and strongly $\aleph_{0}$-separated almost disjoint families because, under the assumption $\mathfrak{p}>\omega_{1}$, a Luzin family is strongly $(\aleph_{0},<\!\mathfrak{c})$-separated and it is not almost normal. This separation property can be further extended in the following natural way:

It turns out that this property is the combinatorical equivalent of the well known topological property, pseudonormality. Recall that a topological space $\mathcal{X}$ is pseudonormal if and only if for every pair of disjoint closed sets, with at least one of them is countable, can be separated by disjoint open sets.

We will stick with the term pseudonormality. The following folklore proposition is already known, as noted in [6]. The proof can be obtained as a corollary of Proposition 2.2. and we write it for completeness.

The reader may notice that follows from the previous proof that in $\Psi(\mathcal{A}_{X})$, for a pair of disjoint closed subsets $F,G$, provided at least one of them is countable:

It is worth noting that by [7, Proposition 3.3], we do have MAD strongly $\aleph_{0}$-separated families. However, the same does not happen for pseudonormal families since, as it is known, in a MAD family we cannot separate an infinite countable set from its complement.

It is straightforward from that definition that strongly $(\aleph_{0},<\mathfrak{c})$-separated almost disjoint families are strongly $\aleph_{0}$-separated and these two definitions are the same under CH. Moreover, it was proved in [7] that almost-normal almost disjoint families are strongly $\aleph_{0}$-separated. It was noticed in [4] that under the assumption that $\mathfrak{p}>\omega_{1}$ there exists a strongly $(\aleph_{0},<\mathfrak{c})$-separated almost disjoint family that is not almost-normal (in fact any Luzin family would have this property).

Recall that in a MAD family $\mathcal{A}$, it is not possible to separate any countable subfamily $\mathcal{B}$ from $\mathcal{A}\setminus\mathcal{B}$. In particular, the existence of an almost-normal MAD family would give an example of almost-normal almost disjoint family that is not strongly $(\aleph_{0},\mathfrak{c})$-separated. This discussion can be summarized by the following diagram:

In the preceding diagram the double arrows are the results that holds in ZFC, the crossed dashed arrows are counter-examples that assumes additional combinatorial axioms and the dotted arrows are implications that remain unknown.

In the previous section we mentioned that in [11] it was proved that $\mathcal{A}$ is potentially normal iff $\mathcal{A}$ does not contain $n$-Luzin gaps. To prove this theorem, the authors of [11] used a forcing notion denoted by $\mathcal{S}_{\mathcal{B},\mathcal{C}}$. Given an almost disjoint family $\mathcal{A}$ and disjoint subsets $\mathcal{B},C$ of $\mathcal{A}$, $\mathcal{S}_{\mathcal{B},\mathcal{C}}$ is the set of all triples $(s,\mathcal{F},\mathcal{G})$ such that $s\in 2^{<\omega}$, $\mathcal{F}\in[\mathcal{B}]^{<\omega}$, $\mathcal{G}\in[\mathcal{C}]^{<\omega}$ and $(\bigcup\mathcal{F})\cap(\bigcup\mathcal{G})\subseteq|s|$. We order $\mathcal{S}_{\mathcal{B},\mathcal{C}}$ by letting $(s,\mathcal{F},\mathcal{G})\leq(s^{\prime},\mathcal{F}^{\prime},\mathcal{G}^{\prime})$ iff $s^{\prime}\subseteq s$, $\mathcal{F}^{\prime}\subseteq\mathcal{F}$, $\mathcal{G}^{\prime}\subseteq\mathcal{G}$, $\forall n\in|s|\setminus|s^{\prime}|(n\in\bigcup\mathcal{F}^{\prime}\rightarrow s(n)=1)$ and $\forall n\in|s|\setminus|s^{\prime}|(n\in\bigcup\mathcal{F}^{\prime}\rightarrow s(n)=0)$. By standard density arguments it is clear that if $G$ is a generic filter then $\bigcup\{s:(s,\mathcal{F},\mathcal{G})\in G\}$ is a characteristic function for a partitioner of $\mathcal{B}\cup\mathcal{C}$ separating $\mathcal{B}$ from $\mathcal{C}$.

In general this poset does not need to preserve cardinals, but it is c.c.c. (in fact, $\sigma$-centered) if either $|\mathcal{B}|=\omega$ or $|\mathcal{C}|=\omega$. So iterating this poset with standard bookkeping techniques we get the following result:

As we have mentioned, this is not true for normality (e.g. Luzin families are not potentially normal, and we have provided an example of a Luzin^∗-family which is not potentially almost-normal).

In this section we introduce the weak $\lambda$-sets, a weakening of the notion of $\lambda$-sets which relate to strongly $\aleph_{0}$-separated almost disjoint families in the same way as $Q$-sets relate to normal almost disjoint families.

We consider $[\omega]^{\omega}$ with the (Polish) topology obtained by identifying $[\omega]^{\omega}$ with the subspace of $2^{\omega}$ corresponding to the characteristic functions of infinite sets. As mentioned in the introduction, if $\mathcal{A}$ is normal (pseudonormal) then $\mathcal{A}$ is a $Q$-set ($\lambda$-set) when viewed as a subspace of $[\omega]^{\omega}$. [9]

Notice that given an almost disjoint family $\mathcal{A}$ and a partitioner $Z$ for $\mathcal{A}$, the sets $\mathcal{B}=\{a\in\mathcal{A}:a\subseteq^{*}Z\}$ and $\mathcal{A}\setminus\mathcal{B}=\{a\in\mathcal{A}:a\cap Z=^{*}\emptyset\}$ are relative $F_{\sigma}$’s and $G_{\delta}$’s of $A$ simultaneously.

Of course, a $\lambda$-set is a weak $\lambda$-set. Moreover, we have the following:

Recall that a set $X$ of reals is a $\lambda^{\prime}$-set if for each countable $A$, $X\cup A$ is a $\lambda$-set. Analogously, we define a set to be a weak $\lambda^{\prime}$-set if for every countable set $A$, $X\cup A$ is a weak $\lambda$ set. It follows easily that weak $\lambda^{\prime}$-sets are weak $\lambda$-sets. Moreover, in the light of the Theorem 4.4, we have the following diagram of implications between special sets of reals:

In [6], forcing was used to construct a consistent example of a $Q$-set $X$ which is concentrated on a countable dense subset $F$ of $2^{\omega}$ ($F$ is in the ground model). The proof actually shows that $X$ is concentrated in every dense subset $F^{\prime}$ of $F$ which is in the ground model. Therefore, by letting $F_{0}$, $F_{1}$ be two disjoint dense subsets of $F$ in the ground model, we get the set $Y=F_{0}\cup F_{1}\cup X$ is perfectly meager and not a weak $\lambda$-set (in the forcing extension). This later fact holds since it is easy to verify that a weak $\lambda$-set cannot be concentrated on two pairwise disjoint countable subsets. This discussion yields the following proposition:

It is still possible that every weak $\lambda$-set is a $\lambda$-set (in ZFC), but it is worth noting that the existence of weak $\lambda$-set that is not a $\lambda$-set cannot be weak $\lambda^{\prime}$.

The next proposition is proved in section 9 of [17].

The following diagram describes the relations we have done so far for almost disjoint families of branches. The double arrows are the results that holds in ZFC, the dashed arrows are implications that assumes additional combinatorial axioms and the dotted arrows are implications that remain unknown. The first line stands for the least size of a set of reals which does not have the respective property. $\mathfrak{q}$ is defined simply as the least size of a set of reals which is not a $Q$-set. For more on $\mathfrak{q}$, see [2]. The fact that $\mathfrak{b}$ is the least size of a non $\lambda$-set and of a non $\sigma$-set is discussed in [16]. The least size of a non perfectly meager set is $\text{non}(\mathcal{M})$ since the least size of a nonmeager set of a Polish space with no isolated points does not depend on the space.

In view of the preceding results, we pose the following:

As discussed in the previous sections, we know that if $\mathcal{A}$ is normal then $\mathcal{A}$ is a $Q$-set (as a subset of $\mathcal{P}(\omega)$). However, the converse is consistently not true, as proved in [13] where a model with a MAD family which is a $Q$-set is constructed. Given the relations between $\sigma$-sets and almost-normality, $\lambda$-sets and pseudonormality and weak $\lambda$-sets and strongly $\aleph_{0}$-separatedness, it is natural to ask if similar results hold for these notions. As we have already discussed, in this sense, strongly $\aleph_{0}$-separated almost disjoint families are weak $\lambda$-sets and pseudonormal almost disjoint families are $\lambda$-sets.

The example from [13] mentioned above shows us that almost disjoint families which are $\lambda$-sets don’t need to be pseudonormal as subsets of $\mathcal{P}(\omega)$ since MAD families are not pseudonormal.

However, the following two questions remain open:

Proposition 2.10 grants the existence of an almost disjoint family of size $\omega_{1}$ which is not almost-normal. Thus, assuming $\mathfrak{b}>\omega_{1}$, we have:

In some sense, there is not much we can do regarding this implication since there are models without $\sigma$-sets ([14], Theorem 22), so in this model “every almost disjoint family which is a $\sigma$-set is almost-normal” is trivially true. Still, we ask:

The diagram below summarizes the known implications and the open questions concerning these properties of $\Psi(A)$ and properties of $A$ as a subset of $[\omega]^{\omega}$.

Looking in a different direction, one may ask about countable paracompactness-like properties of Isbell-Mrówka spaces. It is known that for Hausdorff spaces $X$, $X$ is countably paracompact iff for every decreasing sequence $(F_{n}:n\in\omega)$ of closed subsets of $X$ such that $\bigcap_{n\in\omega}F_{n}=\emptyset$ there exists open sets $(V_{n}:n\in\omega)$ such that $F_{n}\subseteq V_{n}$ and $\bigcap_{n\in\omega}\operatorname{cl}(V_{n})=\emptyset$ (e.g. see [5, Theorem 5.2.1]).