Finite character

In mathematics, a family ${\displaystyle {\mathcal {F}}}$ of sets is of finite character if for each ${\displaystyle A}$, ${\displaystyle A}$ belongs to ${\displaystyle {\mathcal {F}}}$ if and only if every finite subset of ${\displaystyle A}$ belongs to ${\displaystyle {\mathcal {F}}}$. That is,

1. For each ${\displaystyle A\in {\mathcal {F}}}$, every finite subset of ${\displaystyle A}$ belongs to ${\displaystyle {\mathcal {F}}}$.
2. If every finite subset of a given set ${\displaystyle A}$ belongs to ${\displaystyle {\mathcal {F}}}$, then ${\displaystyle A}$ belongs to ${\displaystyle {\mathcal {F}}}$.