# Cardinals associated with an ideal

This is the first in a series of posts leading to a diagram called the Cichon’s diagram. The diagram displays relationships among twelve small cardinals, eight of which are defined by using $\sigma$-ideals. The purpose of this post is to set up the scene.

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

Ideals and $\sigma$-Ideals

Let $X$ be a set. Let $\mathcal{I}$ be a collection of subsets of $X$. We say that $\mathcal{I}$ is an ideal on $X$ if the following three conditions hold.

1. $\varnothing \in \mathcal{I}$.
2. If $A \in \mathcal{I}$ and $B \subset A$, then $B \in \mathcal{I}$.
3. If $A, B \in \mathcal{I}$, then $A \cup B \in \mathcal{I}$.

If $X \notin \mathcal{I}$, then $\mathcal{I}$ is said to be a proper ideal. Note that if $X \in \mathcal{I}$, then $\mathcal{I}$ would simply be the power set of $X$. Thus we would only want to focus on proper ideals. Thus by ideal we mean proper ideal.

We say that $\mathcal{I}$ is a $\sigma$-ideal on $X$ if it is an ideal with the additional property that it is closed under taking countable unions, i.e. if for each $n \in \omega$, $A_n \in \mathcal{I}$, then $\bigcup_{n \in \omega} A_n \in \mathcal{I}$. For the discussion that follows, we even require that all singleton subsets of $X$ are members of any $\sigma$-ideal $\mathcal{I}$.

Elements of an ideal or a $\sigma$-ideal are considered “small sets” or “negligible sets”. The definition of $\sigma$-ideal does indeed reflect how small sets should behave. The empty set is naturally a small set. Any subset of a small set should be a small set. The union of countably many small sets should also be a small set as is any countable set (the union of countably many singleton sets).

Let $\mathcal{B}$ be a collection of subsets of the set $X$. We assume that $\mathcal{B}$ is closed under taking countable unions. It is easy to verify that the set

$\mathcal{I}=\{ A \subset X: \exists \ B \in \mathcal{B} \text{ such that } A \subset B \}$

is a $\sigma$-ideal on $X$. This $\sigma$-ideal $\mathcal{I}$ is said to be generated by the set $\mathcal{B}$. The set $\mathcal{B}$ is called a base for the $\sigma$-ideal $\mathcal{I}$. A subbase for a $\sigma$-ideal is simply a collection of subsets of $X$. Then a base would be generated by taking countable unions of sets in the subbase.

Four Cardinals

We now discuss the cardinal characteristics associated with a $\sigma$-ideal. As before, let $X$ be a set and $\mathcal{I}$ be a $\sigma$-ideal on $X$. As discussed above, we require that all singleton sets are in $\mathcal{I}$. We define the following four cardinals.

$\text{add}(\mathcal{I})=\text{min} \{ \lvert \mathcal{A} \lvert: \mathcal{A} \subset \mathcal{I} \text{ and } \bigcup \mathcal{A} \notin \mathcal{I} \}$

Covering Number
$\text{cov}(\mathcal{I})=\text{min} \{ \lvert \mathcal{A} \lvert: \mathcal{A} \subset \mathcal{I} \text{ and } \bigcup \mathcal{A} = X \}$

Uniformity Number
$\text{non}(\mathcal{I})=\text{min} \{ \lvert A \lvert: A \subset X \text{ and } A \notin \mathcal{I} \}$

Cofinality Number
$\text{cof} (\mathcal{I})=\text{min} \{ \lvert \mathcal{B} \lvert: \mathcal{B} \subset \mathcal{I} \text{ and } \mathcal{B} \text{ is cofinal in } \mathcal{I} \}$

A subset $\mathcal{B}$ of $\mathcal{I}$ is said to be cofinal in $\mathcal{I}$ if for each $A \in \mathcal{I}$, there exists $B \in \mathcal{B}$ such that $A \subset B$, i.e. $\mathcal{B}$ is cofinal in the partial order $\subset$. Such a $\mathcal{B}$ is a base for $\mathcal{I}$.

The numbers $\text{add}(\mathcal{I})$, $\text{cov}(\mathcal{I})$ and $\text{cof} (\mathcal{I})$ are the minimum cardinalities of certain subfamilies of the $\sigma$-ideal $\mathcal{I}$ which fail to be small, i.e. not in $\mathcal{I}$. The additivity number $\text{add}(\mathcal{I})$ is the least cardinality of a subfamily of $\mathcal{I}$ whose union is not in $\mathcal{I}$. The covering number $\text{cov}(\mathcal{I})$ is the minimum cardinality of a subfamily of $\mathcal{I}$ whose union is the entire set $X$. The covering number is the minimum cardinality of a covering of $X$ with elements of $\mathcal{I}$. The cofinality number $\text{cof} (\mathcal{I})$ is the least cardinality of a subfamily of $\mathcal{I}$ that is a cofinal in $\mathcal{I}$. Equivalently, the cofinality number is the least cardinality of a base that generates the $\sigma$-ideal. The uniformity number $\text{non}(\mathcal{I})$ is the least cardinality of a subset of $X$ that is not an element of $\mathcal{I}$.

The elements of the $\sigma$-ideal $\mathcal{I}$ are “small” sets. The additivity number $\text{add}(\mathcal{I})$ is the smallest number of small sets whose union is not small. The covering number $\text{cov}(\mathcal{I})$ is then the smallest number of small sets that fill up the entire set $X$. The uniform number is the least cardinality of a non-small set.

Of these four cardinals, the smallest one is $\text{add}(\mathcal{I})$ and the largest one is $\text{cof} (\mathcal{I})$. Because $\mathcal{I}$ is a $\sigma$-ideal, all four cardinals must be uncountable, hence $\ge \aleph_1$. Obviously $\mathcal{I}$ is confinal in $\mathcal{I}$. Thus $\text{cof} (\mathcal{I}) \le \lvert \mathcal{I} \lvert$. The following inequalities also hold.

$\aleph_1 \le \text{add}(\mathcal{I}) \le \text{cov}(\mathcal{I}) \le \text{cof} (\mathcal{I}) \le \lvert \mathcal{I} \lvert$

$\aleph_1 \le \text{add}(\mathcal{I}) \le \text{non}(\mathcal{I}) \le \text{cof} (\mathcal{I}) \le \lvert \mathcal{I} \lvert$

$\displaystyle \aleph_1 \le \text{add}(\mathcal{I}) \le \text{min} \{ \text{cov}(\mathcal{I}), \text{non}(\mathcal{I}) \} \le \text{max} \{ \text{cov}(\mathcal{I}), \text{non}(\mathcal{I}) \} \le \text{cof} (\mathcal{I}) \le \lvert \mathcal{I} \lvert$

Displaying the Four Cardinals in a Diagram

The inequalities shown in the preceding section can be displayed in a diagram such as the following.

Figure 1 – Cardinal Characteristics of a $\sigma$-Ideal

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

In the above diagram, $a \Rightarrow b$ means $a \le b$. The smallest cardinal $\text{add}(\mathcal{I})$ is lower bounded by $\aleph_1$ on the lower left since $\mathcal{I}$ is a $\sigma$-ideal. The largest cardinal $\text{cof}(\mathcal{I})$ is upper bounded by the cardinality of $\mathcal{I}$ on the upper right since $\mathcal{I}$ is cofinal in $\mathcal{I}$. The diagram tells us that the additivity number is less than or equal to the minimum of the uniformity number and the covering number. On the other hand, the cofinality number is greater than or equal to the maximum of the uniformity number and the covering number.

Examples

In the subsequent posts, we would like to focus on two $\sigma$-ideals, hence eight associated cardinals. To define these two ideals, let $X=\mathbb{R}$, the real line. Let $\mathcal{M}$ be set of all meager subsets of the real line and let $\mathcal{L}$ be the set of all subsets of the real line that are of Lebesgue measure zero.

$\mathcal{M}=\{ A \subset \mathbb{R}: A \text{ is a Meager set} \}$

$\mathcal{L}=\{ A \subset \mathbb{R}: A \text{ is of Lebesgue measure zero} \}$

In the real line, a set is nowhere dense if its closure contains no open set. A meager set is the union of countably many nowhere dense 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, $\mathbb{R} \notin \mathcal{M}$. Thus it is a proper ideal. Similarly, it is straightforward to verify that $\mathcal{L}$ is a $\sigma$-ideal as well as a proper ideal.

Before we examine the ideal $\mathcal{M}$, we consider the $\sigma$-ideal of bounded subsets of $\omega^\omega$ in the next post.

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