# The ideal of meager sets

This is the third in a series of four posts leading to a diagram called The Cichon’s Diagram. This post focuses on the $\sigma$-ideal of meager subsets of the real line. The links to the previous posts: the first post and the second post.

The next post is: the Cichon’s Diagram.

The notion of meager sets can be defined on any topological space. Let $Y$ be a space. A subset $A$ of $Y$ is a nowhere dense set if $\overline{A}$, the closure of $A$ in the space $Y$, contains no open sets. Equivalently, $A$ is a nowhere dense set if for each non-empty open subset $U$ of $Y$, there exists a non-empty open subset $V$ of $U$ such that $V \cap A=\varnothing$. We can always find a part of any open set that misses a nowhere dense set. Thus nowhere dense sets are considered “thin” sets. A subset of $Y$ is said to be a meager set if it is the union of countably many nowhere dense sets. A meager set is also called a set of first category. A non-meager set is then called a set of second category.

Though the notion of meager sets can be considered in any space, we would like to focus on the real line $\mathbb{R}$ or the space of all irrational numbers $\mathbb{P}$. Note that $\mathbb{P}$ is homeomorphic to $\omega^\omega$ (see here). Instead of working with $\mathbb{P}$, we work with $\omega^\omega$, which is the product space of countably many copies of the countable discrete space $\omega$.

$\sigma$-Ideal of Meager Sets

The notion of meager sets is a topological notion of small sets. The real line and the space of irrationals $\omega^\omega$ are “big” sets. This means that they are not the union of countably many meager sets (this fact is a consequence of the Baire category theorem). Let $\mathcal{M}$ be the set of all subsets of the real line that are meager sets. It is straightforward to verify that $\mathcal{M}$ is a $\sigma$-ideal on the real line $\mathbb{R}$. Because of the Baire category theorem, $\mathcal{M}$ is a proper ideal, i.e. $\mathbb{R} \notin \mathcal{M}$. Naturally, we would like to consider the four cardinals associated with this ideal – $\text{add}(\mathcal{M})$ (the additivity number), $\text{cov}(\mathcal{M})$ (the covering number), $\text{non}(\mathcal{M})$ (the uniformity number) and $\text{cof} (\mathcal{M})$ (the cofinality number). These four numbers are displayed in the following diagram.

Figure 1 – Cardinal Characteristics of the $\sigma$-Ideal of Meager Sets

…Cichon…$\displaystyle \begin{array}{ccccccccccccc} \text{ } & \text{ } & \text{ } &\text{ } & \bold n \bold o \bold n ( \mathcal{M} ) &\text{ } &\Longrightarrow &\text{ } &\bold c \bold o \bold f (\mathcal{M} )&\text{ } &\Longrightarrow & \text{ } & 2^{\aleph_0}\\ \text{ } & \text{ } & \text{ } &\text{ } & \text{ } &\text{ } &\text{ } &\text{ } &\text{ }&\text{ } &\text{ } & \text{ } & \text{ } \\ \text{ } & \text{ } & \text{ } &\text{ } & \Uparrow &\text{ } &\text{ } &\text{ } &\Uparrow&\text{ } &\text{ } & \text{ }& \text{ } \\ \text{ }& \text{ } & \text{ } &\text{ } & \text{ } &\text{ } &\text{ } &\text{ } &\text{ }&\text{ } &\text{ } & \text{ }& \text{ } \\ \aleph_1 & \text{ } & \Longrightarrow & \text{ } &\bold a \bold d \bold d ( \mathcal{M} ) & \text{ } &\Longrightarrow & \text{ } &\bold c \bold o \bold v ( \mathcal{M} ) &\text{ } &\text{ } &\text{ } & \text{ } \end{array}$

In the above diagram, an arrow means $\le$. So $\alpha \Rightarrow \beta$ means $\alpha \le \beta$. The inequalities displayed in this diagram always hold for any $\sigma$-ideal. The only inequality that requires explanation is $\text{cof} (\mathcal{M}) \le 2^{\aleph_0}$. Any meager set is a subset of an $F_\sigma$-set. To see this, let $A=\bigcup_{n \in \omega} X_n$ where each $X_n$ is a nowhere dense subset of the real line. Then $A \subset \bigcup_{n \in \omega} \overline{X_n}$. Each $\overline{X_n}$ is also nowhere dense. Thus the set of all $F_\sigma$ nowhere dense sets is cofinal in $\mathcal{M}$. This cofinal set has cardinality continuum. Of course, if continuum hypothesis holds ($\aleph_1=2^{\aleph_0}$), then all four cardinals are identical and are $\aleph_1$.

$\sigma$-Ideal of Bounded Sets

In some respects, it is more advantageous to consider the $\sigma$-ideal of meager subsets of $\mathbb{P}$, the set of all irrational numbers, or equivalently $\omega^\omega$. Thus we consider the $\sigma$-ideal of meager subsets of $\omega^\omega$. We also use $\mathcal{M}$ denote this $\sigma$-ideal. Note that the calculation of the four cardinals $\text{add}(\mathcal{M})$, $\text{cov}(\mathcal{M})$, $\text{non}(\mathcal{M})$ and $\text{cof} (\mathcal{M})$ yields the same values regardless of whether $\mathcal{M}$ is the $\sigma$-ideal of meager subsets of the real line or of $\omega^\omega$. In the remainder of this post, $\mathcal{M}$ is the $\sigma$-ideal of meager subsets of $\omega^\omega$, the space of the irrational numbers.

Let $\mathcal{S}$ be the collection of all $\sigma$-compact subsets of $\omega^\omega$. In this previous post, the following $\sigma$-ideal is discussed.

$\mathcal{K}=\{ A \subset \omega^\omega: \exists \ B \in \mathcal{S} \text{ such that } A \subset B \}$

It is straightforward to verify that $\mathcal{K}$ is indeed a $\sigma$-ideal on $\omega^\omega$. This is what we know about this $\sigma$-ideal from this previous post.

• $A \in \mathcal{K}$ if and only if $A$ is a bounded subset of $\omega^\omega$.
• $\mathfrak{b}=\text{add}(\mathcal{K})=\text{non}(\mathcal{K})$.
• $\mathfrak{d}=\text{cov}(\mathcal{K})=\text{cof}(\mathcal{K})$.

So the sets in $\mathcal{K}$ are simply the bounded sets. For this $\sigma$-ideal, the additivity number and the uniformity numbers are $\mathfrak{b}$, the bounding number. The covering number and the cofinality number are the dominating number $\mathfrak{d}$. As we will see below, these facts provide insight on the $\sigma$-ideal $\mathcal{M}$.

The following lemma connects the $\sigma$-ideal $\mathcal{K}$ with the $\sigma$-ideal $\mathcal{M}$.

Lemma 1
Let $A$ be a compact subset of $\omega^\omega$. Then $A$ is a closed and nowhere dense subset of $\omega^\omega$. Hence any $\sigma$-compact subset of $\omega^\omega$ is a meager subset of $\omega^\omega$.

Proof of Lemma 1
Since $A$ is compact, for each $n$, the projection of $A$ into the $n$th factor of $\omega^\omega$ is compact and thus finite. Let $[0, g(n)]=\{ j \in \omega: 0 \le j \le g(n) \}$ be a finite set that contains the $n$th projection of $A$. Thus $A \subset \prod_{n \in \omega} [0, g(n)]$. It is straightforward to verify that $\prod_{n \in \omega} [0, g(n)]$ is nowhere dense in $\omega^\omega$. Thus $A$ is a closed nowhere dense subset of $\omega^\omega$. It follows that any $\sigma$-compact subset of $\omega^\omega$ is a meager subset of $\omega^\omega$. $\square$

Theorem 2
As a result of Lemma 1, we have $\mathcal{K} \subset \mathcal{M}$. However, $\mathcal{M} \not \subset \mathcal{K}$. Thus the two $\sigma$-ideals are not the same.

Any example that proves Theorem 2 would be an unbounded meager set. One such example is constructed in this previous post.

More on $\sigma$-Ideal of Meager Sets

For $\mathcal{K}$, the $\sigma$-ideal generated by $\sigma$-compact subsets of $\omega^\omega$, and for $\mathcal{M}$, the $\sigma$-ideal of meager sets in $\omega^\omega$, we are interested in the four associated cardinals add, non, cov and cof in each $\sigma$-ideal. For $\mathcal{K}$, the four cardinals are just two, $\mathfrak{b}$ and $\mathfrak{d}$. We would like to relate these six cardinals, plus $\aleph_1$ and $2^{\aleph_0}$. They are represented in the following diagram.

Figure 5 – Partial Cichon’s Diagram

…Cichon…$\displaystyle \begin{array}{ccccccccccccc} \text{ } & \text{ } & \text{ } &\text{ } & \bold n \bold o \bold n ( \mathcal{M} ) &\text{ } &\Longrightarrow &\text{ } &\bold c \bold o \bold f (\mathcal{M} )&\text{ } &\Longrightarrow & \text{ } & 2^{\aleph_0}\\ \text{ } & \text{ } & \text{ } &\text{ } & \text{ } &\text{ } &\text{ } &\text{ } &\text{ }&\text{ } &\text{ } & \text{ } & \text{ }\\ \text{ }& \text{ } & \text{ } &\text{ } & \Uparrow &\text{ } &\text{ } &\text{ } &\Uparrow &\text{ } &\text{ } & \text{ } & \text{ }\\ \text{ } & \text{ } & \text{ } &\text{ } & \text{ } &\text{ } &\text{ } &\text{ } &\text{ }&\text{ } &\text{ } & \text{ }& \text{ } \\ \text{ } & \text{ } & \text{ } &\text{ } & \mathfrak{b} &\text{ } &\Longrightarrow &\text{ } &\mathfrak{d}&\text{ } &\text{ } & \text{ }& \text{ } \\ \text{ }& \text{ } & \text{ } &\text{ } & \text{ } &\text{ } &\text{ } &\text{ } &\text{ }&\text{ } &\text{ } & \text{ }& \text{ } \\ \text{ } & \text{ } & \text{ } &\text{ } & \Uparrow &\text{ } &\text{ } &\text{ } &\Uparrow &\text{ } &\text{ } & \text{ } & \text{ }\\ \text{ } & \text{ } & \text{ } &\text{ } & \text{ } &\text{ } &\text{ } &\text{ } &\text{ }&\text{ } &\text{ } & \text{ }& \text{ } \\ \aleph_1 & \text{ } & \Longrightarrow & \text{ } &\bold a \bold d \bold d ( \mathcal{M} ) & \text{ } &\Longrightarrow & \text{ } &\bold c \bold o \bold v ( \mathcal{M} ) &\text{ } &\text{ } &\text{ } & \text{ } \end{array}$

As in the other diagrams, arrows mean $\le$. So $\alpha \Longrightarrow \beta$ means $\alpha \le \beta$. Furthermore, we have two additional relations.

…Cichon…$\text{add}(\mathcal{M})=\text{min}(\mathfrak{b},\ \text{cov}(\mathcal{M}))$
…Cichon…$\text{cof}(\mathcal{M})=\text{max}(\mathfrak{d},\ \text{non}(\mathcal{M}))$

As shown, Figure 5 is not complete. It only has information on the $\sigma$-ideal $\mathcal{M}$ on meager sets. The usual Cichon’s diagram would also include the four associated cardinals for $\mathcal{L}$, the $\sigma$-ideal of Lebesgue measure zero sets. In this post we focus on $\mathcal{M}$. The full Cichon’s diagram will be covered in a subsequent post. insert

In Figure 5, the cardinals go from smaller to the larger from left to right and bottom to top. It starts with $\aleph_1$ on the lower left and moves toward the continuum on the upper right. Because of Theorem 3, the cardinals associated with the $\sigma$-ideal $\mathcal{K}$ are represented by $\mathfrak{b}$ and $\mathfrak{d}$ in the diagram. We next examine the inequalities between the cardinals associated with $\mathcal{M}$ and $\mathfrak{b}$ and $\mathfrak{d}$.

There are four inequalities to account for. First, $\mathfrak{b} \le \text{non}(\mathcal{M})$ and $\text{cov}(\mathcal{M}) \le \mathfrak{d}$. The first inequality follows from the fact that $\mathfrak{b} = \text{non}(\mathcal{K})$ and that $\mathcal{K} \subset \mathcal{M}$. The second inequality follows from the fact that $\mathfrak{d} = \text{cov}(\mathcal{K})$ and that $\mathcal{K} \subset \mathcal{M}$.

The inequality $\text{add}(\mathcal{K}) \le \mathfrak{b}$ follows from the fact that if $F \subset \omega^\omega$ is an unbounded set, then there exist $\lvert F \lvert$ many meager sets whose union is a non-meager set. This fact is established in this previous post (see Theorem 1 in that post).

For the inequality $\mathfrak{d} \le \text{cof}(\mathcal{M})$, see Corollary 5.4 of [1]. For the additional inequalities, see Theorem 5.6 in [1].

The next post is on the full Cichon’s Diagram.

Reference

1. Blass, A., Combinatorial Cardinal Characteristics of the Continuum, Handbook of Set Theory (M. Foreman, A. Kanamori, eds), Springer Science+Business Media B. V., Netherlands, 395-489, 2010.

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