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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 $ℐ$ is an ideal of sets such that $\cup ℐ=X\notin ℐ$, write add($ℐ\right)=min\left\{#\left(ℰ\right):$ $ℰ\subseteq ℐ$, $\cup ℰ\notin ℐ\right\}$; non($ℐ\right)=min\left\{#\left(A\right):$ $A\subseteq X$, $A\notin ℐ\right\}$; cov($ℐ\right)=min\left\{#\left(ℰ\right):$ $ℰ\subseteq ℐ$, $\cup ℰ=X\right\}$; cf($ℐ\right)=min\left\{#\left(ℰ\right):$ $ℰ\subseteq ℐ$, $ℰ$ is cofinal with $ℐ\right\}$. Write $𝒩$ for the ideal of Lebesgue negligible subsets of $ℝ$, $ℳ$ for the ideal of meagre subsets of $ℝ$, $𝒦$ for the $\sigma$-ideal of subsets of ${ℕ}^{ℕ}$ generated by the compact sets. We find that non($𝒦\right)=add\left(𝒦\right)$ and that cov($𝒦\right)=cf\left(𝒦\right)$; call these cardinals $𝔟$ and $𝔡$ respectively. All what is known about the cardinals associated with $𝒩$, $ℳ$ and $𝒦$ 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: ${\omega }_{1}\le add\left(𝒩\right)\le add\left(ℳ\right)\le 𝔟\le 𝔡\le cf\left(ℳ\right)\le cf\left(𝒩\right)\le 𝔠$, but cov($𝒩\right)>𝔟$ and cov($𝒩\right)<𝔟$ are both relatively consistent with ZFC. Further: add($ℳ\right)=min\left(𝔟,cov\left(ℳ\right)\right)$, cf($ℳ\right)=max\left(𝔡,non\left(ℳ\right)\right)$. I give full proofs of the results establishing the mentioned diagram and references to some of the relevant consistency results.
##### MSC:
 03E05 Combinatorial set theory (logic) 03E10 Ordinal and cardinal arithmetic 28A05 Classes of sets 03E35 Consistency; independence results (set theory)