family of sets

A **family of sets** consists of an **index set** $I$ and, for each element $k$ of $I$, a set $S_k$.

Given $I$, a set $F$ and a function $p\colon F \to I$, we get a family of sets by defining $S_k$ to be the preimage $p^*(k)$.

Conversely, given a family of sets, let $F$ be the disjoint union

$\biguplus_k S_k = \{ (k,x) | k \in I, x \in S_k \}$

and let $f(k,x)$ be $k$.

(We should talk about ways to formalise this concept in various forms of set theory and when the latter construction above requires the axiom of collection.)

Last revised on January 15, 2011 at 06:00:41. See the history of this page for a list of all contributions to it.