Solution
= Solution
\b[Not a <topology>.] For every integer $k\geq2$, the set $U_k=\mathbb Z\setminus k\mathbb Z$ is periodic with period $k$, so belongs to the family. Their union is $\mathbb Z\setminus\{0\}$: for a nonzero $n$, choose $k>|n|$, which cannot divide $n$. But this union has no positive integer period. If it had period $k$, membership of $-k$ would force membership of zero. Again arbitrary unions fail the <topology> axiom.