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 nth 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 nth 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.

Additional Relationships

…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.

\text{ }

\text{ }

\text{ }

Dan Ma topology

Daniel Ma topology

Dan Ma math

Daniel Ma mathematics

\copyright 2020 – Dan Ma

3 thoughts on “The ideal of meager sets

  1. Pingback: The Cichon’s Diagram | Dan Ma's Topology Blog

  2. Pingback: The ideal of bounded sets | Dan Ma's Topology Blog

  3. Pingback: Cardinals associated with an ideal | Dan Ma's Topology Blog

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s