Subset of Countable Set is Countable
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A subset of a countable set is countable.
Let $S$ be a countable set.
Let $T \subseteq S$.
By definition, there exists an injection $f: S \to \N$.
Let $i: T \to S$ be the inclusion mapping.
We have that $i$ is an injection.
Because the composite of injections is an injection, it follows that $f \circ i: T \to \N$ is an injection.
Hence, $T$ is countable.
- Subset of Countably Infinite Set is Countable, a special case of this theorem
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Exercise $17.12$
- 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 7$: Countable and Uncountable Sets: Corollary $7.3$