Definition:Subfamily

Definition
Let $I$ and $S$ be sets.

Let $x: I \to S$ be an indexing function for $S$.

Let $\family {x_i}_{i \mathop \in I}$ be a family indexed by $I$.

Let $J \subseteq I$ be a subset of $I$.

The family $\family {x_j}_{j \mathop \in J}$ indexed by $J$ is known as a subfamily of $\family {x_i}_{i \mathop \in I}$.