Add, cov, non and cof for null and meagre ideals

Work in ZFC. Let $\mathbb{R}$ carry its usual topology and its Lebesgue measure $\lambda$ (Open subset of $\mathbb{R}$ (every point has a neighbourhood inside it), closed subset (complement open), and clopen, The $\epsilon$-neighbourhood and the punctured $\epsilon$-neighbourhood of a point of $\mathbb{R}$, Lebesgue measurable sets, the family $\mathcal{L}({\mathbb{R}}^n)$, and the restricted set function ${\lambda }_n$), and let a subset of $\mathbb{R}$ be meagre when it is a union of a sequence of nowhere dense sets (Nowhere dense, meagre, residual, and comeagre subsets of a topological space; this is the same class as the one of Nowhere dense, meager (first category), residual, and second category subsets of $\mathbb{R}$, which requires the displayed union to equal the set, because subsets of nowhere dense sets are nowhere dense). The two families of the title are

$\mathcal{N}:=\{E\subseteq \mathbb{R}:E\text{ is Lebesgue measurable and }\lambda (E)=0\},\mathcal{M}:=\{E\subseteq \mathbb{R}:E\text{ is meagre}\},$

the Lebesgue-null ideal and the meagre ideal on the real line (Measure-null sets and almost-everywhere statements relative to a measure). Since $(\mathbb{R},\mathcal{L}(\mathbb{R}),\lambda )$ is complete (Assuming countable choice, $\mathcal{L}({\mathbb{R}}^n)$ is a sigma-algebra containing every elementary set and ${\lambda }_n$ is a complete measure extending elementary volume), a set belongs to $\mathcal{N}$ exactly when it is contained in a Lebesgue-null measurable set, so the measurability clause in the definition of $\mathcal{N}$ is not an extra restriction on the members.

The four invariants. Let $X$ be a set and let $\mathcal{I}$ be a family of subsets of $X$ (in the applications below $\mathcal{I}$ is $\mathcal{N}$ or $\mathcal{M}$ and $X=\mathbb{R}$). Define

• the additivity $\mathrm{add}(\mathcal{I}):=\min \{\mid \mathcal{A}\mid :\mathcal{A}\subseteq \mathcal{I}\text{ and }\bigcup \mathcal{A}\mathrm{\notin }\mathcal{I}\}$;
• the covering number $\mathrm{cov}(\mathcal{I}):=\min \{\mid \mathcal{A}\mid :\mathcal{A}\subseteq \mathcal{I}\text{ and }\bigcup \mathcal{A}=X\}$;
• the non number $\mathrm{non}(\mathcal{I}):=\min \{\mid Y\mid :Y\subseteq X\text{ and }Y\mathrm{\notin }\mathcal{I}\}$;
• the cofinality $\mathrm{cof}(\mathcal{I}):=\min \{\mid \mathcal{A}\mid :\mathcal{A}\subseteq \mathcal{I}\text{ and }\mathrm{\forall }B\in \mathcal{I}\mathrm{\exists }A\in \mathcal{A}(B\subseteq A)\}$.

The family $\mathcal{A}$ in the last clause is an inclusion-cofinal subfamily of $\mathcal{I}$; "I-cover" is the reading of the second clause when $\mathcal{I}$ is an ideal of subsets of $X$.

The four minima exist and are attained for each of $\mathcal{N}$ and $\mathcal{M}$, with $X=\mathbb{R}$. For the two "family" numbers one exhibits a single family: the family of singletons $\mathcal{S}:=\{\{x\}:x\in \mathbb{R}\}$, indexed by $\mathbb{R}$, is a subfamily both of $\mathcal{N}$ and of $\mathcal{M}$. Each singleton is null, indeed has $\lambda (\{x\})=0$ (Every at most countable subset of ${\mathbb{R}}^n$ is Lebesgue null; in particular ${\lambda }_1(\mathbb{Q})=0$, where the Axiom of Countable Choice The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$) supplies the measure theory), and each singleton is meagre: $\mathbb{R}\setminus \{x\}$ is open, because for $y\mathrm{\ne }x$ the neighbourhood $N_{\delta }(y)$ with $\delta =\mid y-x\mid \mathrm{/}2>0$ contains no $z$ with $z=x$, since $z=x$ would give $\mid z-y\mid =\mid x-y\mid =2\delta ≮\delta$ (The $\epsilon$-neighbourhood and the punctured $\epsilon$-neighbourhood of a point of $\mathbb{R}$, Open subset of $\mathbb{R}$ (every point has a neighbourhood inside it), closed subset (complement open), and clopen, Ordered field); so $\{x\}$ is closed, it has empty interior, since $N_{\epsilon }(x)\subseteq \{x\}$ would put $x+\epsilon \mathrm{/}2\mathrm{\ne }x$ into $\{x\}$, and therefore $\{x\}$ is nowhere dense (Interior, closure, boundary and exterior of a subset of $\mathbb{R}$), hence meagre, its union with the constant sequence of empty sets being $\{x\}$ itself. Since $\bigcup \mathcal{S}=\mathbb{R}$, the families of candidates for $\mathrm{add}$ and for $\mathrm{cov}$ are nonempty and contain the cardinality $\mid \mathcal{S}\mid$. For the remaining two numbers a single set suffices: $\mathbb{R}$ itself is neither null nor meagre, because $\lambda (\mathbb{R})=+\mathrm{\infty }\mathrm{\ne }0$ (Assuming countable choice, $\mathcal{L}({\mathbb{R}}^n)$ is a sigma-algebra containing every elementary set and ${\lambda }_n$ is a complete measure extending elementary volume) and no meagre subset of $\mathbb{R}$ exhausts $\mathbb{R}$ (Baire category in $\mathbb{R}$, by nested intervals with canonically chosen rational endpoints: a countable intersection of dense open sets is dense, so $\mathbb{R}$ is not a countable union of nowhere dense sets), so $\mathbb{R}$ witnesses $Y\subseteq X$, $Y\mathrm{\notin }\mathcal{N}$ and $Y\mathrm{\notin }\mathcal{M}$; and $\mathcal{I}$ itself is an inclusion-cofinal subfamily of $\mathcal{I}$, since $B\subseteq B$ for $B\in \mathcal{I}$.

Each of the four candidate collections is therefore a nonempty set of ordinals, so each has a least element: cardinalities are available, and are cardinals, because the Axiom of Choice well-orders every set (The Axiom of Choice, A set equinumerous with some ordinal has a least such ordinal, that ordinal is a cardinal, and equinumerous sets get the same one; no choice principle is used), and the candidate collection is the image under $Z\mapsto \mid Z\mid$ of a subset of the power set of $\mathcal{I}$ or of $X$, hence a set by Replacement. The minimum is attained: there is a subfamily of $\mathcal{I}$ of size $\mathrm{add}(\mathcal{I})$ whose union is not in $\mathcal{I}$, a subfamily of $\mathcal{I}$ of size $\mathrm{cov}(\mathcal{I})$ with union $X$, a set $Y\subseteq X$ of size $\mathrm{non}(\mathcal{I})$ with $Y\mathrm{\notin }\mathcal{I}$, and an inclusion-cofinal subfamily of $\mathcal{I}$ of size $\mathrm{cof}(\mathcal{I})$. All four numbers are cardinals (A set equinumerous with some ordinal has a least such ordinal, that ordinal is a cardinal, and equinumerous sets get the same one; no choice principle is used), and no further property is built into the definition: the elementary inequalities among the eight numbers, and their comparison with ${\mathrm{\aleph }}_1$ and $\mathfrak{c}=2^{{\mathrm{\aleph }}_0}$, are proved in Elementary bounds on ideal cardinal invariants (Cardinal sum $\kappa \oplus \lambda$, product $\kappa \otimes \lambda$ and exponentiation ${\kappa }^{\lambda }$, and why they are written apart from the ordinal operations).

Conventions. Bartoszyński's list of cardinal invariants of an ideal $\mathcal{J}$ of subsets of a set $X$ is exactly the four displayed clauses: he writes $\mathrm{add}(\mathcal{J})=\min \{\mid \mathcal{A}\mid :\mathcal{A}\subseteq \mathcal{J}$ and $\bigcup \mathcal{A}\mathrm{\notin }\mathcal{J}\}$, $\mathrm{cov}(\mathcal{J})=\min \{\mid \mathcal{A}\mid :\mathcal{A}\subseteq \mathcal{J}$ and $\bigcup \mathcal{A}=X\}$, $\mathrm{non}(\mathcal{J})=\min \{\mid Y\mid :Y\subseteq X$ and $Y\mathrm{\notin }\mathcal{J}\}$ and $\mathrm{cof}(\mathcal{J})=\min \{\mid \mathcal{A}\mid :\mathcal{A}\subseteq \mathcal{J}$ and $\mathrm{\forall }B\in \mathcal{J}\mathrm{\exists }A\in \mathcal{A}(B\subseteq A)\}$. The eight numbers of this page are the values of the four functions at $\mathcal{I}=\mathcal{N}$ and $\mathcal{I}=\mathcal{M}$, read as $\mathrm{add}(\mathcal{N})$, $\mathrm{cov}(\mathcal{N})$, $\mathrm{non}(\mathcal{N})$, $\mathrm{cof}(\mathcal{N})$, $\mathrm{add}(\mathcal{M})$, $\mathrm{cov}(\mathcal{M})$, $\mathrm{non}(\mathcal{M})$ and $\mathrm{cof}(\mathcal{M})$. It is part of the definition that these are evaluated on the real line, with Lebesgue measure and the usual topology; $\mathcal{N}$ and $\mathcal{M}$ are proper (that is, $\mathbb{R}\mathrm{\notin }\mathcal{I}$) and, under the Axiom of Countable Choice, $\sigma$-ideals, by Null sets are closed under countable unions and, in a complete space, under arbitrary subsets and The meagre subsets of a topological space form a sigma-ideal, but neither closure property is used in the definition.

Choice accounting. The minima use AC via A set equinumerous with some ordinal has a least such ordinal, that ordinal is a cardinal, and equinumerous sets get the same one; no choice principle is used, exactly as the sibling definitions of $p$, $t$, $b$, $d$, $s$ and $r$ do. The null side uses ${\mathrm{A}\mathrm{C}}_{\omega }$ through the cited suppliers (Assuming countable choice, $\mathcal{L}({\mathbb{R}}^n)$ is a sigma-algebra containing every elementary set and ${\lambda }_n$ is a complete measure extending elementary volume, Every at most countable subset of ${\mathbb{R}}^n$ is Lebesgue null; in particular ${\lambda }_1(\mathbb{Q})=0$); the meagre side uses no choice principle to speak of meagreness or in the two facts needed above — the two elementary computations for $\{x\}$ and the Baire fact for $\mathbb{R}$ (Baire category in $\mathbb{R}$, by nested intervals with canonically chosen rational endpoints: a countable intersection of dense open sets is dense, so $\mathbb{R}$ is not a countable union of nowhere dense sets).

The four numbers were introduced by the descriptive-set-theory school in the context of an ideal of subsets of a Polish space, and Bartoszyński's chapter opens with precisely this list; the transfer of the definitions between the real line and Cantor space $2^{\omega }$ is developed below, together with the comparison of the eight values on the two spaces (Transfer of null and meagre invariants between Cantor space and the line), and the Cichoń diagram collecting the inequalities among them is The ZFC inequalities of Cichoń's diagram.

The names are mnemonics rather than descriptions of the definitions: the "additivity" is the least size of a subfamily whose union escapes the ideal, not the additivity of a measure-like functional; the "covering number" counts covers by ideal members rather than general covers; "non" counts the least size of a set not in the ideal; and the "cofinality" is computed in the inclusion order of the ideal, not in the order of the underlying set. The clause "$\bigcup \mathcal{A}\mathrm{\notin }\mathcal{I}$" in the definition of $\mathrm{add}$ excludes the empty family when $\mathrm{\varnothing }\in \mathcal{I}$. The equation $\bigcup \mathcal{A}=X$ excludes the empty family when $X\mathrm{\ne }\mathrm{\varnothing }$; if $X=\mathrm{\varnothing }$, the empty family instead witnesses $\mathrm{cov}(\mathcal{I})=0$. Here $X=\mathbb{R}\mathrm{\ne }\mathrm{\varnothing }$ and both ideals contain $\mathrm{\varnothing }$, so the empty subfamily is never a candidate for either additivity or covering.