Recently, I have read Brendle's article *Between p-points and nowhere dense ultrafilters* [Isr. J. Math. 113, 205-230]. In this paper, he noted that $\mathrm{cof}(\mathcal{C},\mathcal{M}) = \mathrm{cov}(\mathcal{M})$, where $\mathcal{C}$ represents the set of all closed countable subsets of the real line. But I don't know how to prove this; a proof sketch would be appreciated.


I also want to know about $\mathrm{cof}(\mathcal{C}',\mathcal{M})$, where $\mathcal{C}'$ represents the set of all countable subsets of the real line.



A proof for cof(C,M)=cov(M)
Let's work on Baire space $\omega^\omega$. $\mathrm{cof}(\mathcal{C},\mathcal{M})$ is the least size of a collection of meager sets so that any countable closed set is a subset of the member of the collection. $\mathrm{cof}(\mathcal{C}',\mathcal{M})$ is the same, but where $\mathcal{C}'$ is all possible countable sets. $\mathrm{cov}(\mathcal{M})$ is the least size of a collection of meager sets whose union equals $\omega^\omega$. 


I claim that all three numbers are equal. In other words, in terms of cardinality it doesn't matter whether we ask for a collection of meager sets cofinal over the countable sets, the closed countable sets, or even just the one element sets.


$\mathrm{cof}(\mathcal{C},\mathcal{M})\leq \mathrm{cof}(\mathcal{C}',\mathcal{M})$ is straight from the definition.


$\mathrm{cov}(\mathcal{M})\leq\mathrm{cof}(\mathcal{C},\mathcal{M})$ is almost as straightforward; since sets of size 1 are closed any collection of meager sets cofinal in the countable closed sets covers the whole space.


$\mathrm{cof}(\mathcal{C}',\mathcal{M})\leq \mathrm{cov}(\mathcal{M})$: We use a characterization of $\mathrm{cov}(\mathcal{M})$ due to Bartoszynski (this is Theorem 2.4.1 in his book with Judah): it is the least size of a family $\mathcal{F}\subseteq\omega^\omega$ so that any $g\in\omega^\omega$ is eventually different from some $f\in\mathcal{F}$, meaning $(\forall^\infty n)f(n)\not=g(n)$. So fix such a family. Also fix a partition of $\omega$ into countably many infinite pieces $\{A\_k:k\in\omega\}$. For each $f\in\mathcal{F}$ let $M\_f$ be the set of all $x\in\omega^\omega$ for which there exists a $k$ so that $(\forall^\infty n\in A\_k)f(n)\not=x(n)$. Then $M\_f$ is meager. We claim the collection of $M\_f$ is cofinal over the countable sets.


Fix $C=\{h\_k:k\in\omega\}$ a countable set. Let $g\_C\in\omega^\omega$ be given by $g\_C(n)=h\_k(n)$ if $n\in A\_k$. There is some $f\in\mathcal{F}$ so that $g\_C$ is eventually different from $f$. And this implies that $C\subseteq M\_f$. QED.


