The Cichon’s Diagram

The Cichon’s Diagram is a diagram that shows the relationships among ten small cardinals – four cardinals associated with the \sigma-ideal of sets of Lebesgue measure zero, four cardinals associated with the \sigma-ideal of sets of meager sets, the bounding number \mathfrak{b}, and the dominating number \mathfrak{d}. What makes this interesting is that elements of analysis, topology and set theory flow into the same spot. Here’s the diagram.

Figure 1 – The Cichon’s Diagram

In this diagram, \alpha \rightarrow \beta means \alpha \le \beta. The preceding three posts (the first post, the second post and the third post) give the necessary definitions and background to understand the diagram. In addition to the above diagram, the following relationships also hold.

Figure 2 – The Cichon’s Diagram – Additional Relationships

The Cardinal Characteristics of a \sigma -Ideal

For any \sigma-ideal \mathcal{I} on a set X, there are four associated cardinals – \text{add}(\mathcal{I}), \text{non}(\mathcal{I}), \text{cov}(\mathcal{I}) and \text{cof}(\mathcal{I}). The first one is the additivity number, which is the least number of elements of \mathcal{I} whose union is not an element of \mathcal{I}. The second cardinal is called the uniformity number, which is the least cardinality of a subset of X that is not an element of \mathcal{I}. The third cardinal is called the covering number, which is the least cardinality of a subfamily of \mathcal{I} that is also a covering of X. The fourth cardinal is called the cofinality number, which is the least cardinality of a subfamily of \mathcal{I} that is cofinal in \mathcal{I}. For more information, see the first post. The four cardinals are related in a way that is depicted in the following diagram. Again, \alpha \Rightarrow \beta means \alpha \le \beta.

Figure 3 – 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}

Figure 3 explains the basic orientation of the Cichon’s Diagram. Filling it with three \sigma-ideals produces the Cichon’s Diagram.

The Three \sigma -Ideals in the Cichon’s Diagram

Let \mathcal{K} be the \sigma-ideal of bounded subsets of \omega^\omega. It is known that \mathfrak{b}=\text{add}(\mathcal{K})=\text{non}(\mathcal{K}) (this is called the bounding number) and \mathfrak{d}=\text{cov}(\mathcal{K})=\text{cof}(\mathcal{K}) (this is called the dominating number). The ideal \mathcal{K} is discussed in this previous post. Let \mathcal{M} be the \sigma-ideal of meager subsets of the real line \mathbb{R} (this is discussed in this previous post). Let \mathcal{L} be the \sigma-ideal of Lebesgue measure zero subsets of the real line.

Thus the Cichon’s Diagram (Figure 1 above) houses information about three \sigma-ideals. The two numbers for the \sigma-ideal \mathcal{K} are situated in the middle of the diagram (\mathfrak{b} and \mathfrak{d}). The four numbers for the \sigma-ideal \mathcal{M} are situated in the center portion of the diagram. The four numbers for the \sigma-ideal \mathcal{L} are located on the left side and the right side. The Cichon’s Diagram (Figure 1) is flanked by \aleph_1 on the lower left and by continnum 2^{\aleph_0} on the upper right.

More on the Cichon’s Diagram

One interesting aspect of the Cichon’s Diagram: it is a small diagram with small cardinals where elements of analysis (measure) and topology (category) come together. The following diagram shows the path that includes both the bounding number and the dominating number.

Figure 4 – The Cichon’s Diagram – The Main Path

The path circled in the above diagram involves all three \sigma-ideals. It is also one of the longest increasing paths in the diagram.

    \aleph_1 \le \text{add}(\mathcal{L}) \le \text{add}(\mathcal{M}) \le \mathfrak{b} \le \mathfrak{d} \le \text{cof}(\mathcal{M}) \le \text{cof}(\mathcal{L}) \le 2^{\aleph_0}

There are fifteen arrows in Figure 1. The proofs of these arrows (or inequalities) require varying degrees of effort. Three are basic information – \aleph_1 \le \text{add}(\mathcal{L}), \mathfrak{b} \le \mathfrak{d} and \text{cof}(\mathcal{L}) \le 2^{\aleph_0}. Because \mathcal{L} is a \sigma-ideal, its additivity number must be uncountable. By definition, \mathfrak{b} \le \mathfrak{d}. The \sigma-ideal \mathcal{L} has a cofinal subfamily consisting of Borel sets. Thus \text{cof}(\mathcal{L}) \le 2^{\aleph_0}.

Four of the arrows follow from the relative magnitude of the four cardinals of a \sigma-ideal as shown in Figure 3 – \text{add}(\mathcal{L}) \le \text{cov}(\mathcal{L}), \text{non}(\mathcal{L}) \le \text{cof}(\mathcal{L}), \text{add}(\mathcal{M}) \le \text{cov}(\mathcal{M}) and \text{non}(\mathcal{M}) \le \text{cof}(\mathcal{M}).

Three of the arrows are proved in this previous post\mathfrak{b} \le \text{non}(\mathcal{M}), \mathfrak{d} \le \text{cov}(\mathcal{M}) and \text{add}(\mathcal{M}) \le \mathfrak{b}. The last inequality follows from this fact: if F \subset \omega^\omega is an unbounded set, then there exist \lvert F \lvert many meager subsets of the real line whose union is a non-meager set, essentially a result in Miller [8].

The proofs of the remaining five arrows can be found in [3] – \mathfrak{d} \le \text{cof}(\mathcal{M}), \text{add}(\mathcal{L}) \le \text{add}(\mathcal{M}), \text{cov}(\mathcal{L}) \le \text{non}(\mathcal{M}), \text{cof}(\mathcal{M}) \le \text{cof}(\mathcal{L}) and \text{cov}(\mathcal{M}) \le \text{non}(\mathcal{L}). The proofs of two additional relationships displayed in Figure 2 can also be found in [3].

The fifteen arrows in the Cichon’s Diagram represent the only inequalities among the ten cardinals (not counting \aleph_1 and 2^{\aleph_0}) that are provable in ZFC [1] and [5]. As illustration, we give an example of non-ZFC provable relation in the next section.

An Example of an Inequality Not Provable in ZFC

In the following diagram, the cardinals \mathfrak{b} and \text{cov}(\mathcal{M}) are encircled. These two numbers are not connected by arrows.

Figure 5 – The Cichon’s Diagram – An Example of Non-ZFC Provable

We sketch out a proof that no inequalities can be established between \mathfrak{b} and \text{cov}(\mathcal{M}). First Martin’s Axiom (MA) implies that \mathfrak{b} \le \text{cov}(\mathcal{M}). Topologically, the statement MA (\kappa) means that any compact Hausdorff space X that satisfies the countable chain condition cannot be the union of \kappa or fewer many nowhere dense sets. The Martin’s Axiom (MA) is the statement that MA (\kappa) holds for all \kappa less than 2^{\aleph_0}. It follows that MA implies that \text{cov}(\mathcal{M}) cannot be less than 2^{\aleph_0} and thus \text{cov}(\mathcal{M})=2^{\aleph_0}. It is always the case that the bounding number \mathfrak{b} is \le 2^{\aleph_0}.

On the other hand, in Laver’s model [6] for the Borel conjecture, \mathfrak{b} > \text{cov}(\mathcal{M}). In Laver’s model, every subset of the real line that is of strong measure zero is countable. Since any set with the Rothberger property is of strong measure zero, every subset of the real line that has the Rothberger property is countable in Laver’s model. Let \text{non}(\text{Rothberger}) be the least cardinality of a subset of the real line that does not have the Rothberger property. Thus in Laver’s model, \text{non}(\text{Rothberger})=\aleph_1. It is well known that \text{non}(\text{Rothberger})=\text{cov}(\mathcal{M}); see Theorem 5 in [10]. Thus in Laver’s model, \text{cov}(\mathcal{M})=\aleph_1.

In Laver’s model, \mathfrak{b} > \aleph_1. Note that \mathfrak{b}= \aleph_1 implies that there is an uncountable subset of the real line that is concentrated about \mathbb{Q}, the set of all rational numbers; see Theorem 10.2 in [12]. Any concentrated set is of strong measure zero; see Theorem 3.1 in [9]. Thus it must be the case that \mathfrak{b} > \aleph_1=\text{cov}(\mathcal{M}) in Laver’s model.

Remarks

The Cichon’s Diagram is a remarkable diagram. It blends elements of analysis and topology into a small diagram. The fifteen arrows shown in the diagram are obviously far from the end of the story. The Cichon’s Diagram had been around for a long time. Much had been written about it. The article [13] posted some questions about the diagram. See [1], [2], [4] and [11] for further information on the cardinals in the diagram.

Reference

  1. Bartoszynski, T., Judah H., Shelah S.,The Cichon Diagram, J. Symbolic Logic, 58(2), 401-423, 1993.
  2. Bartoszynski, T., Judah H., Shelah S.,Set theory: On the structure of the real line, A K
    Peters, Ltd.. Wellesley, MA, 1995.
  3. 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.
  4. Fremlin, D. H., Cichon’s diagram. In Seminaire d’Initiation ´a l’Analyse, 23, Universite Pierre et Marie Curie, Paris, 1984.
  5. Garcia, H., da Silva S. G., Identifying Small with Bounded: Unboundedness, Domination, Ideals and Their Cardinal Invariants, South American Journal of Logic, 2 (2), 425-436, 2016.
  6. Laver, R., On the consistency of Borel’s conjecture, Acta Math., 137, 151-169, 1976.
  7. Miller, A. W., Some Properties of Measure and Category, Trans. Amer. Math. Soc., 266 (1), 93-114, 1981.
  8. Miller, A. W., A Characterization of the Least Cardinal for Which the Baire Category Theorem Fails, Proc. Amer. Math. Soc., 86 (3), 498-502, 1982.
  9. Miller, A. W., Special Subsets of the Real Line, Handbook of Set-Theoretic Topology (K. Kunen and J. E. Vaughan, eds), Elsevier Science Publishers B. V., Amsterdam, 201-233, 1984.
  10. Miller A. W., Fremlin D. H., On some properties of Hurewicz, Menger, and Rothberger, Fund. Math., 129, 17-33, 1988.
  11. Pawlikowski, J., Reclaw I., Parametrized Cichon’s diagram and small sets, Fund. Math., 127, 225-239, 1987.
  12. Van Douwen, E. K., The Integers and Topology, Handbook of Set-Theoretic Topology (K. Kunen and J. E. Vaughan, eds), Elsevier Science Publishers B. V., Amsterdam, 111-167, 1984.
  13. Vaughn, J. E., Small uncountable cardinals and topology, Open Problems in Topology (J. van Mill and G.M. Reed, eds), Elsevier Science Publishers B.V. (North-Holland), 1990.

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3 thoughts on “The Cichon’s Diagram

  1. Pingback: The ideal of meager sets | Dan Ma's Topology Blog

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

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

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