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For fixed positive integer period $n$, integer-valued periodic functions on $\mathbb{Z}$ are in bijection with ${\mathbb{Z}}^n$ and hence form a countable set. Their union over periods is a countable union of countable sets. In contrast, period-one integer-valued functions on $\mathbb{Q}$ can prescribe arbitrary binary values on its infinitely many distinct classes modulo $\mathbb{Z}$. The power set of a countably infinite set injects into them, proving uncountability.

If every sequence of distinct elements of a positive-real set $A$ has arithmetic averages tending to zero, then $A$ is a countable set. For each positive integer $n$, the threshold set $\{a\in A:a\ge 1/n\}$ must be finite: an infinite sequence inside it would have averages at least $1/n$. The Archimedean property makes $A$ their union, so the countable union of countable sets theorem applies. The hypothesis concerns every distinct-element sequence, not merely one enumeration.

Every family of pairwise disjoint nonempty open disks in the plane is countable. The density of the rational numbers gives each disk a point of the countable set ${\mathbb{Q}}^2$. Fixing an enumeration and taking the first point in each disk yields an injection from disks to labels. This argument requires no countable choice of representatives and is useful when geometric separation would otherwise produce an uncountable family.

A forcing order has the countable chain condition when every antichain in a forcing order is countable. Such forcing preserves cardinals and cofinalities at least ${\aleph }_1$.

A countable ordinal has a countable set as its underlying set.

Let $T$ be an ergodic transformation preserving a probability measure $\mu$ on a countable set. Some state $x$ has $\mu (\{x\})>0$. Along its forward orbit,An infinite sequence of distinct states with masses bounded below by $\mu (\{x\})$ is impossible, so the orbit eventually reaches a finite cycle $Y$. Since $Y\subseteq T^{-1}Y$ and invariance gives $\mu (T^{-1}Y)=\mu (Y)>0$, the difference is null. The cycle is invariant modulo null sets, and ergodicity therefore gives $\mu (Y)=1$.

A finite Cartesian product of countable sets is countable. For a fixed finite exponent $m$, index tuples in ${\mathbb{N}}^m$ can be listed by increasing maximum coordinate, with each group finite. If $B$ is countably infinite and $m>0$, its power $B^m$ is also countably infinite.

The first uncountable ordinal has cofinality ${\omega }_1$. Indeed, the supremum of any countable set of countable ordinals is still a countable ordinal, so no countable sequence is cofinal in ${\omega }_1$.

A set is hereditarily countable when its transitive closure is countable. These sets form the transitive set $H_{{\omega }_1}$ and all have rank below ${\omega }_1$.