**Setup**




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Let $X$ be a set of cardinality $\kappa\geq \aleph\_0$.


*Edit:*


Based on **Todd Eisworth**'s suggestion:


What is the minimum cardinality of a collection $\hat{X}$ of countable subsets of $X$ such that every countable subset $A \subseteq X$ is contained in some element of $\hat{X}$?




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Minimum cardinality of a cofinal collection of countable subsets of a set
As it is stated in the comments, the question is about the cofinality of $([\lambda]^{\aleph\_0}, \subseteq)$.


The following definition is due to Shelah:


$
cov(\lambda, \mu, \theta, \sigma)=min\{|P|: P$ is a family of subsets of $\lambda$ each of size $< \mu$ such that for every $a \subseteq \lambda, |a|<\theta$, for some $\alpha < \sigma$ and $A\_i \in P,$ for $i<\alpha,$ we have $a \subseteq \bigcup\_{i<\alpha}A\_i \}$.


It is easily sean that


1-$cf([\lambda]^{\aleph\_0}, \subseteq)=cov(\lambda, \aleph\_1, \aleph\_1, 2)$.


By a theorem of Shelah (see [Sh: 355], $\aleph \_{\omega +1}$ has a Jonsson Algebra, see also [Analytic guides and updates for cardinal arithmetic](https://shelah.logic.at/v1/E12.pdf) page 23)


2-$\lambda^{\aleph\_0}=cov(\lambda, \aleph\_1, \aleph\_1, 2)+ 2^{\aleph\_0}$.


Note that in particular, if $\lambda^{\aleph\_0} > 2^{\aleph\_0}$, then


$cf([\lambda]^{\aleph\_0}, \subseteq)=\lambda^{\aleph\_0}$.


