# The ideal of bounded sets

This is the second in a series of posts leading to a diagram called The Cichon’s Diagram. In this post, we examine an ideal that will provide insight on the ideal of meager sets, which is part of the Cichon’s Diagram. For the definitions of ideal and $\sigma$-ideal, see the first post.

The next two posts are: the third post and the fourth post – the Cichon’s Diagram.

Let $\omega$ be the set of all non-negative integers, i.e. $\omega=\{ 0,1,2,\cdots \}$. Let $X=\omega^\omega$, the set of all functions from $\omega$ into $\omega$. We can also think of $X$ as a topological space since it is a product space of countably many copies of the discrete space $\omega$. As a product space, $X=\omega^\omega$ is homeomorphic to $\mathbb{P}$, the space of all irrational numbers with the usual real line topology (see here).

Recall that for $f,g \in \omega^\omega$, $f \le^* g$ means that $f(n) \le g(n)$ for all but finitely many $n$. This is a partial order that is called the eventual domination order. A subset $F$ of $\omega^\omega$ is a bounded set if there is a $g \in \omega^\omega$ such that $g$ is an upper bound of $F$ with respect to the partial order $\le^*$, i.e. for each $f \in F$, we have $f \le^* g$. The set $F$ is an unbounded set of it is not bounded. The set $F$ is a dominating set if for each $g \in \omega^\omega$, there exists $f \in F$ such that $g \le^* f$, i.e. the set $F$ is cofinal in $\omega^\omega$ with respect to the eventual domination order $\le^*$.

We are interested in the least cardinality of an unbounded set and the least cardinality of a dominating set. The former is denoted by $\mathfrak{b}$ and is called the bounding number while the latter is denoted by $\mathfrak{d}$ and is called the dominating number.

An Interim Ideal

We define two ideals on $X=\omega^\omega$. Let $\mathcal{S}$ be the collection of all $\sigma$-compact subsets of $\omega^\omega$.

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

$\mathcal{K}_b=\{ A \subset \omega^\omega: A \text{ is a bounded set} \}$

The first one $\mathcal{K}_\sigma$ is the set of all subsets of $\omega^\omega$, each of which is contained in a $\sigma$-compact set. The second one $\mathcal{K}_b$ is simply the set of all bounded subsets. It is straightforward to verify that $\mathcal{K}_\sigma$ is a $\sigma$-ideal on $\omega^\omega$. Note that any countable set $\{ f_0, f_1,f_2,\cdots \} \subset \omega^\omega$ is a bounded set (via a diagonal argument). Thus the union of countably many bounded sets $A_0,A_1,A_2,\cdots$ with $A_n$ having an upper bound $f_n$ must be a bounded set. The $f_n$ have an upper bound $f$, which is an upper bound of the union of the sets $A_n$. Thus $\mathcal{K}_b$ is a $\sigma$-ideal on $\omega^\omega$.

Furthermore, since $\omega^\omega$ is not $\sigma$-compact, $\mathcal{K}_\sigma$ is a proper ideal. Likewise $\omega^\omega$ is an unbounded set, $\mathcal{K}_b$ is a proper ideal. The ideal $\mathcal{K}_\sigma$ is called the $\sigma$-ideal generated by $\sigma$-compact subsets of $\omega^\omega$. The ideal $\mathcal{K}_b$ is the $\sigma$-ideal of bounded subset of $\omega^\omega$. However, these two ideals are one and the same.

Theorem 1
Let $F \subset \omega^\omega$. Then the following conditions are equivalent.

1. The set $F$ is bounded.
2. There exists a $\sigma$-compact set $X$ such that $F \subset X \subset \omega^\omega$.
3. With $F$ as a subset of the real line, the set $F$ is an $F_\sigma$-subset of $F \cup \mathbb{Q}$ where $\mathbb{Q}$ is the set of all rational numbers.

Theorem 1 is the Theorem 1 found in
. The sets satisfying Condition 1 of this theorem are precisely the elements of the $\sigma$-ideal $\mathcal{K}_b$. The sets satisfying Condition 2 of this theorem are precisely the elements of the $\sigma$-ideal $\mathcal{K}_\sigma$. According to this theorem, the two $\sigma$-ideals are the same. Each is a different characterization of the same $\sigma$-ideal. As a result, we drop the subscript and call this $\sigma$-ideal $\mathcal{K}$.

Four Cardinals

With the $\sigma$-ideal $\mathcal{K}$ from the preceding section, we would like to examine the four associated cardinals $\text{add}(\mathcal{K})$ (the additivity number), $\text{non}(\mathcal{K})$ (the uniformity number), $\text{cov}(\mathcal{K})$ (the covering number) and $\text{cof}(\mathcal{K})$ (the cofinality number). For the definitions of these numbers, see the first post.

Figure 1 – Cardinal Characteristics of the $\sigma$-Ideal Generated by $\sigma$-Compact Sets

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

In the diagram, $\alpha \Rightarrow \beta$ means that $\alpha \le \beta$. The additivity number $\text{add}(\mathcal{K})$ is lowered bounded by $\aleph_1$ on the lower right in the diagram since the ideal $\mathcal{K}$ is a $\sigma$-ideal. The middle of the diagram shows the relationships that hold for any $\sigma$-ideal. To see that $\text{cof}(\mathcal{K}) \le 2^{\aleph_0}$, define $B_f=\{ h \in \omega^\omega: h \le^* f \}$ for each $f \in \omega^\omega$. The set of all $B_f$ is cofinal in $\mathcal{K}$. The inequality holds since there are $2^{\aleph_0}$ many sets $B_f$.

We can further refine Figure 1. The following theorem shows how.

Theorem 2
The values of the four cardinals associated with the $\sigma$-ideal $\mathcal{K}$ are the bounding numbers $\mathfrak{b}$ and the dominating number $\mathfrak{d}$. Specifically, we have the following equalities.

$\mathfrak{b}=\text{add}(\mathcal{K})=\text{non}(\mathcal{K})$
$\mathfrak{d}=\text{cov}(\mathcal{K})=\text{cof}(\mathcal{K})$

Proof of Theorem 2
Based on the discussion in the first post, $\text{add}(\mathcal{K}) \le \text{non}(\mathcal{K})$ and $\text{cov}(\mathcal{K}) \le \text{cof}(\mathcal{K})$ always hold. We establish the equalities by showing the following.

$\mathfrak{b} \le \text{add}(\mathcal{K}) \le \text{non}(\mathcal{K}) = \mathfrak{b}$
$\mathfrak{d} \le \text{cov}(\mathcal{K}) \le \text{cof}(\mathcal{K}) \le \mathfrak{d}$

Viewing $\mathcal{K}$ as a $\sigma$-ideal of bounded sets, $\text{non}(\mathcal{K})$ is the least cardinality of an unbounded set. Thus $\mathfrak{b}=\text{non}(\mathcal{K})$.

To see $\mathfrak{b} \le \text{add}(\mathcal{K})$, let $\mathcal{A} \subset \mathcal{K}$ such that $\lvert \mathcal{A} \lvert=\text{add}(\mathcal{K})$ and $Y=\bigcup \mathcal{A} \notin \mathcal{K}$. Note that each $A \in \mathcal{A}$ is a bounded set with an upper bound $f(A) \in \omega^\omega$. We claim that $F=\{ f(A): A \in \mathcal{A} \}$ is unbounded. This is because $Y=\bigcup \mathcal{A}$ is unbounded. Since there exists an unbounded set $F$ with cardinality $\text{add}(\mathcal{K})$, it follows that $\mathfrak{b} \le \text{add}(\mathcal{K})$.

To see $\text{cof}(\mathcal{K}) \le \mathfrak{d}$, let $F \subset \omega^\omega$ be a dominating set such that $\lvert F \lvert=\mathfrak{d}$. Note that for each $f \in \omega^\omega$, the set $B_f=\{ h \in \omega^\omega: h \le^* f \}$ is a bounded set and thus $B_f \in \mathcal{K}$. It can be verified that $\mathcal{B}=\{ B_f: f \in F \}$ is cofinal in $\mathcal{K}$. Since there is a cofinal set $\mathcal{B}$ with cardinality $\mathfrak{d}$, it follows that $\text{cof}(\mathcal{K}) \le \mathfrak{d}$.

To see $\mathfrak{d} \le \text{cov}(\mathcal{K})$, let $\mathcal{W} \subset \mathcal{K}$ such that $\lvert \mathcal{W} \lvert=\text{cov}(\mathcal{K})$ and $\bigcup \mathcal{W}=\omega^\omega$. For each $A \in \mathcal{W}$, let $f(A)$ be an upper bound of $A$. It can be verified that the set $F=\{ f(A): A \in \mathcal{W} \}$ is a dominating set. Since we have a dominating set $F$ with cardinality $\text{cov}(\mathcal{K})$, we have $\mathfrak{d} \le \text{cov}(\mathcal{K})$. This completes the proof of Theorem 2. $\square$

With additional information from Theorem 2, Figure 1 can be revised as follows:

Figure 2 – Revised Figure 1

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

Note that there are only four cardinals in this diagram – $\aleph_1$, $\mathfrak{b}$, $\mathfrak{d}$ and $2^{\aleph_0}$. Of course, if continuum hypothesis holds, there would only one number in the diagram, namely $\aleph_1$.

The next post is on the $\sigma$-ideal $\mathcal{M}$ of meager sets.

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