Redundant finite character family definition Definition: Let $F$ be a family of sets. $F$ is called of finite character if for each set $A$, we have that:
$A\in F\iff$ Each finite subset of $A$, is also in $F$.
I can't see the point of this definition, I find it redundant.
It seems trivial, if $A$ is in $F$ then every subset (finite or infinite) of $A$ will be always in $F$.
My thinking is that every family $F$ is of finite character, according to the definition.

As Somos pointed out in the comments, this is not "redundant". Let's consider the vector space $\mathbb{R}^{\alpha}$ for some dimension (possible infinite) $\alpha \ge 2$ and let $F_1$ be the collection of all linearly independent subsets $A \subseteq \mathbb R^{\alpha}$. $F_1$ is of finite character since a subset $A \subseteq \mathbb R^{\alpha}$ is linearly independent if and only if every finite subset $A^* \subseteq A$ is linearly independent over $\mathbb R^{\alpha}$.
Now consider $F_2$ - the collection of all maximal linearly independent subsets $A \subseteq \mathbb R^{\alpha}$. $F_2$ is not of finite character since for any maximal linearly independent $A \in F_2$ and any $a \in A$ we may consider the finite subset $\{a\} \subseteq A$. Since $\alpha \ge 2$ the set $\{a \}$ is no maximal linearly independent over $\mathbb R^{\alpha}$ and hence $\{a\} \not \in F_2$.
