Definition 4.4.1 (Elementary Family).label Let $X$ be a set and $\cali{E}\suf 2^{X}$, then $\cali{E}$ is an elementary family if:
- (P1)
$\emp\in\cali{E}$
- (P2)
For any $E,F\in\cali{E},E\cap F\in\cali{E}$
- (E)
For any $E,F\in\cali{E}$ with $E\suf F$, there exists $\curl{E_i}_{1}^{n}\suf\cali{E}$ such that $F\del E=\dcups{n}{i=1}E_{i}$
If $X\in\cali{E}$ then (E) is equivalent to
- (E’)
For any $E\in\cali{E}$, there exists $\curl{E_i}_{1}^{n}\suf\cali{E}$ such that $E^{c}=\dcups{n}{i=1}E_{i}$
Proof. Suppose (E’) holds. Let $E,F\in\cali{E}$ with $E\suf F$ and pick $\curl{E_i}_{1}^{n}\suf\cali{E}$ such that $E^{c}=\dcups{n}{i=1}E_{i}$ then $F\del E=F\cap E^{c}=\dcups{n}{i=1}F\cap E_{i}$ and $F\cap E_{i}\in \cali{E}$ for each $i$ by (P2)$\square$
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