Countable set

A countable set is a set $A$ for which there exists an injective function $f:A\to\mathbb{N}$, where $\mathbb{N}$ is the set of natural numbers . Equivalently, $A$ is countable if it is finite or there exists a bijective function between $A$ and $\mathbb{N}$.

Countability is a statement about cardinality : a set is countable exactly when its size is no larger than that of $\mathbb{N}$. Many constructions in analysis and topology distinguish sharply between countable and uncountable sets.

Examples:

• The integers $\mathbb{Z}$ form a countable set (they can be arranged in a sequence).
• The rational numbers $\mathbb{Q}$ form a countable set.