Intersection of Family is Subset of Intersection of Subset of Family
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Theorem
Let $I$ be an indexing set.
Let $\family {A_\alpha}_{\alpha \mathop \in I}$ be an indexed family of subsets of a set $S$.
Let $J \subseteq I$.
Then:
- $\ds \bigcap_{\alpha \mathop \in I} A_\alpha \subseteq \bigcap_{\alpha \mathop \in J} A_\alpha$
where $\ds \bigcap_{\alpha \mathop \in I} A_\alpha$ denotes the intersection of $\family {A_\alpha}_{\alpha \mathop \in I}$.
Proof
\(\ds x\) | \(\in\) | \(\ds \bigcap_{\alpha \mathop \in I} A_\alpha\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \forall \alpha \in I: \, \) | \(\ds x\) | \(\in\) | \(\ds A_\alpha\) | Intersection is Subset | |||||||||
\(\ds \leadsto \ \ \) | \(\ds \forall \alpha \in J: \, \) | \(\ds x\) | \(\in\) | \(\ds A_\alpha\) | Definition of Subset: $J \subseteq I$ | |||||||||
\(\ds \leadsto \ \ \) | \(\ds x\) | \(\in\) | \(\ds \bigcap_{\alpha \mathop \in J} A_\alpha\) | Definition of Intersection of Family |
$\blacksquare$
Also see
Sources
- 1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Subsets and Complements; Union and Intersection: Theorem $3 \ \text{(a)}$
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 4$: Indexed Families of Sets: Exercise $3 \ \text{(a)}$