This is the third in a series of four posts leading to a diagram called The Cichon’s Diagram. This post focuses on the -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 be a space. A subset of is a nowhere dense set if , the closure of in the space , contains no open sets. Equivalently, is a nowhere dense set if for each non-empty open subset of , there exists a non-empty open subset of such that . 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 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 or the space of all irrational numbers . Note that is homeomorphic to (see here). Instead of working with , we work with , which is the product space of countably many copies of the countable discrete space .
-Ideal of Meager Sets
The notion of meager sets is a topological notion of small sets. The real line and the space of irrationals 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 be the set of all subsets of the real line that are meager sets. It is straightforward to verify that is a -ideal on the real line . Because of the Baire category theorem, is a proper ideal, i.e. . Naturally, we would like to consider the four cardinals associated with this ideal – (the additivity number), (the covering number), (the uniformity number) and (the cofinality number). These four numbers are displayed in the following diagram.
Figure 1 – Cardinal Characteristics of the -Ideal of Meager Sets
In the above diagram, an arrow means . So means . The inequalities displayed in this diagram always hold for any -ideal. The only inequality that requires explanation is . Any meager set is a subset of an -set. To see this, let where each is a nowhere dense subset of the real line. Then . Each is also nowhere dense. Thus the set of all nowhere dense sets is cofinal in . This cofinal set has cardinality continuum. Of course, if continuum hypothesis holds (), then all four cardinals are identical and are .
-Ideal of Bounded Sets
In some respects, it is more advantageous to consider the -ideal of meager subsets of , the set of all irrational numbers, or equivalently . Thus we consider the -ideal of meager subsets of . We also use denote this -ideal. Note that the calculation of the four cardinals , , and yields the same values regardless of whether is the -ideal of meager subsets of the real line or of . In the remainder of this post, is the -ideal of meager subsets of , the space of the irrational numbers.
Let be the collection of all -compact subsets of . In this previous post, the following -ideal is discussed.
It is straightforward to verify that is indeed a -ideal on . This is what we know about this -ideal from this previous post.
- if and only if is a bounded subset of .
So the sets in are simply the bounded sets. For this -ideal, the additivity number and the uniformity numbers are , the bounding number. The covering number and the cofinality number are the dominating number . As we will see below, these facts provide insight on the -ideal .
The following lemma connects the -ideal with the -ideal .
Let be a compact subset of . Then is a closed and nowhere dense subset of . Hence any -compact subset of is a meager subset of .
Proof of Lemma 1
Since is compact, for each , the projection of into the th factor of is compact and thus finite. Let be a finite set that contains the th projection of . Thus . It is straightforward to verify that is nowhere dense in . Thus is a closed nowhere dense subset of . It follows that any -compact subset of is a meager subset of .
As a result of Lemma 1, we have . However, . Thus the two -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 -Ideal of Meager Sets
For , the -ideal generated by -compact subsets of , and for , the -ideal of meager sets in , we are interested in the four associated cardinals add, non, cov and cof in each -ideal. For , the four cardinals are just two, and . We would like to relate these six cardinals, plus and . They are represented in the following diagram.
Figure 5 – Partial Cichon’s Diagram
As in the other diagrams, arrows mean . So means . Furthermore, we have two additional relations.
As shown, Figure 5 is not complete. It only has information on the -ideal on meager sets. The usual Cichon’s diagram would also include the four associated cardinals for , the -ideal of Lebesgue measure zero sets. In this post we focus on . 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 on the lower left and moves toward the continuum on the upper right. Because of Theorem 3, the cardinals associated with the -ideal are represented by and in the diagram. We next examine the inequalities between the cardinals associated with and and .
There are four inequalities to account for. First, and . The first inequality follows from the fact that and that . The second inequality follows from the fact that and that .
The inequality follows from the fact that if is an unbounded set, then there exist 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 , see Corollary 5.4 of . For the additional inequalities, see Theorem 5.6 in .
The next post is on the full Cichon’s Diagram.
- 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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