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 -ideal, see the first post.
Let be the set of all non-negative integers, i.e. . Let , the set of all functions from into . We can also think of as a topological space since it is a product space of countably many copies of the discrete space . As a product space, is homeomorphic to , the space of all irrational numbers with the usual real line topology (see here).
Recall that for , means that for all but finitely many . This is a partial order that is called the eventual domination order. A subset of is a bounded set if there is a such that is an upper bound of with respect to the partial order , i.e. for each , we have . The set is an unbounded set of it is not bounded. The set is a dominating set if for each , there exists such that , i.e. the set is cofinal in with respect to the eventual domination order .
We are interested in the least cardinality of an unbounded set and the least cardinality of a dominating set. The former is denoted by and is called the bounding number while the latter is denoted by and is called the dominating number.
An Interim Ideal
We define two ideals on . Let be the collection of all -compact subsets of .
The first one is the set of all subsets of , each of which is contained in a -compact set. The second one is simply the set of all bounded subsets. It is straightforward to verify that is a -ideal on . Note that any countable set is a bounded set (via a diagonal argument). Thus the union of countably many bounded sets with having an upper bound must be a bounded set. The have an upper bound , which is an upper bound of the union of the sets . Thus is a -ideal on .
Furthermore, since is not -compact, is a proper ideal. Likewise is an unbounded set, is a proper ideal. The ideal is called the -ideal generated by -compact subsets of . The ideal is the -ideal of bounded subset of . However, these two ideals are one and the same.
Let . Then the following conditions are equivalent.
- The set is bounded.
- There exists a -compact set such that .
- With as a subset of the real line, the set is an -subset of where 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 -ideal . The sets satisfying Condition 2 of this theorem are precisely the elements of the -ideal . According to this theorem, the two -ideals are the same. Each is a different characterization of the same -ideal. As a result, we drop the subscript and call this -ideal .
With the -ideal from the preceding section, we would like to examine the four associated cardinals (the additivity number), (the uniformity number), (the covering number) and (the cofinality number). For the definitions of these numbers, see the first post.
Figure 1 – Cardinal Characteristics of the -Ideal Generated by -Compact Sets
In the diagram, means that . The additivity number is lowered bounded by on the lower right in the diagram since the ideal is a -ideal. The middle of the diagram shows the relationships that hold for any -ideal. To see that , define for each . The set of all is cofinal in . The inequality holds since there are many sets .
We can further refine Figure 1. The following theorem shows how.
The values of the four cardinals associated with the -ideal are the bounding numbers and the dominating number . Specifically, we have the following equalities.
Proof of Theorem 2
Based on the discussion in the first post, and always hold. We establish the equalities by showing the following.
Viewing as a -ideal of bounded sets, is the least cardinality of an unbounded set. Thus .
To see , let such that and . Note that each is a bounded set with an upper bound . We claim that is unbounded. This is because is unbounded. Since there exists an unbounded set with cardinality , it follows that .
To see , let be a dominating set such that . Note that for each , the set is a bounded set and thus . It can be verified that is cofinal in . Since there is a cofinal set with cardinality , it follows that .
To see , let such that and . For each , let be an upper bound of . It can be verified that the set is a dominating set. Since we have a dominating set with cardinality , we have . This completes the proof of Theorem 2.
With additional information from Theorem 2, Figure 1 can be revised as follows:
Figure 2 – Revised Figure 1
Note that there are only four cardinals in this diagram – , , and . Of course, if continuum hypothesis holds, there would only one number in the diagram, namely .
The next post is on the -ideal of meager sets.
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