Let's say a family of sets, $F$ has finite character iff $\forall X: [X \in F \leftrightarrow \forall E \subseteq X: E \, finite \rightarrow E \in F)$
or more precise: let's say a family $F$ has character k iff $\forall X: X \in F \leftrightarrow (\forall E \subseteq X: |E| \leq k \rightarrow E \in F)$.
Now I'm wondering, what would be a sensible definition for a family to be of (at most) countable infinite character?)
And can you think of an example for a family of countable infinite character, but not finite character?
EDIT: Would something like $\forall X: X \in F \leftrightarrow (\forall E \subseteq X: |E| \leq |\omega| \rightarrow E \in F)$ be legitimate? (whereby $\omega$ is the smallest inductive set, i.e. the set-theory-description of $\mathbb{N_0}$)
EDIT: Would the power set of $\mathbb{N}$ be such an example and if so, how could we write this without using $\mathbb{N}$? And would there be a maximal element in it?