## Cichoń’s diagram.(English)Zbl 0559.03029

Publ. Math. Univ. Pierre Marie Curie 66, Sémin. Initiation Anal. 23ème Année-1983/84, Exp. No. 5, 13 p. (1984).
In this note I collect results of F. Rothberger, A. W. Miller, T. Bartoszyński, J. Raisonnier and J. Stern on the relationships between ten cardinal numbers lying between $$\omega_ 1$$ and the continuum. If $${\mathcal I}$$ is an ideal of sets such that $$\cup {\mathcal I}=X\not\in {\mathcal I}$$, write add($${\mathcal I})=\min \{\#({\mathcal E}):$$ $${\mathcal E}\subseteq {\mathcal I}$$, $$\cup {\mathcal E}\not\in {\mathcal I}\}$$; non($${\mathcal I})=\min \{\#(A):$$ $$A\subseteq X$$, $$A\not\in {\mathcal I}\}$$; cov($${\mathcal I})=\min \{\#({\mathcal E}):$$ $${\mathcal E}\subseteq {\mathcal I}$$, $$\cup {\mathcal E}=X\}$$; cf($${\mathcal I})=\min \{\#({\mathcal E}):$$ $${\mathcal E}\subseteq {\mathcal I}$$, $${\mathcal E}$$ is cofinal with $${\mathcal I}\}$$. Write $${\mathcal N}$$ for the ideal of Lebesgue negligible subsets of $${\mathbb{R}}$$, $${\mathcal M}$$ for the ideal of meagre subsets of $${\mathbb{R}}$$, $${\mathcal K}$$ for the $$\sigma$$-ideal of subsets of $${\mathbb{N}}^{{\mathbb{N}}}$$ generated by the compact sets. We find that non($${\mathcal K})=add({\mathcal K})$$ and that cov($${\mathcal K})=cf({\mathcal K})$$; call these cardinals $${\mathfrak b}$$ and $${\mathfrak d}$$ respectively. All what is known about the cardinals associated with $${\mathcal N}$$, $${\mathcal M}$$ and $${\mathcal K}$$ is captured by a diagram, developed by J. Cichoń, A. Kamburelis and J. Pawlikowski [Proc. Am. Math. Soc. 94, 142-146 (1985)]. In detail, the following is known: $$\omega_ 1\leq add({\mathcal N})\leq add({\mathcal M})\leq {\mathfrak b}\leq {\mathfrak d}\leq cf({\mathcal M})\leq cf({\mathcal N})\leq {\mathfrak c}$$, but cov($${\mathcal N})>{\mathfrak b}$$ and cov($${\mathcal N})<{\mathfrak b}$$ are both relatively consistent with ZFC. Further: add($${\mathcal M})=\min ({\mathfrak b},cov({\mathcal M}))$$, cf($${\mathcal M})=\max ({\mathfrak d},non({\mathcal M}))$$. I give full proofs of the results establishing the mentioned diagram and references to some of the relevant consistency results.

### MSC:

 03E05 Other combinatorial set theory 03E10 Ordinal and cardinal numbers 28A05 Classes of sets (Borel fields, $$\sigma$$-rings, etc.), measurable sets, Suslin sets, analytic sets 03E35 Consistency and independence results