Intersection is Subset/Family of Sets
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Theorem
Let $\family {S_\alpha}_{\alpha \mathop \in I}$ be a family of sets indexed by $I$.
Then:
- $\ds \forall \beta \in I: \bigcap_{\alpha \mathop \in I} S_\alpha \subseteq S_\beta$
where $\ds \bigcap_{\alpha \mathop \in I} S_\alpha$ is the intersection of $\family {S_\alpha}_{\alpha \mathop \in I}$.
Proof
\(\ds x\) | \(\in\) | \(\ds \bigcap_{\alpha \mathop \in I} S_\alpha\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \forall \beta \in I: \, \) | \(\ds x\) | \(\in\) | \(\ds S_\beta\) | Definition of Intersection of Family | |||||||||
\(\ds \leadsto \ \ \) | \(\ds \forall \beta \in I: \, \) | \(\ds \bigcap_{\alpha \mathop \in I} S_\alpha\) | \(\subseteq\) | \(\ds S_\beta\) | Definition of Subset |
$\blacksquare$
Sources
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 4$: Indexed Families of Sets: Exercise $1 \ \text{(b)}$