Basic Properties of Countable Sets

Let $\mathbb{N}$ be the set of natural numbers, and let countable be understood as in the definition of a countable set. Then the following hold.

1. $\mathbb{N}$ is countable.

2. Every finite set is countable.

3. (Subsets.) If $X$ is countable and $Y\subseteq X$, then $Y$ is countable.

4. (Images.) If $X$ is countable, $Y$ is a set and $g:X\to Y$ is a map with $Y=\{g(x):x\in X\}$, then $Y$ is countable.

5. (Injections.) If $Y$ is countable and $h:X\to Y$ is a map such that $h(x)=h(x')$ implies $x=x'$ for all $x,x'\in X$, then $X$ is countable.

6. (Adjoining a point.) If $S$ is a set, $X\subseteq S$ is countable and $z\in S$, then $X\cup\{z\}$ is countable.