Indexed set family/Multiples of n/Example

For ${}n\in \mathbb {N} _{+}$, let

${}\mathbb {N} n={\left\{x\in \mathbb {N} _{+}\mid x{\text{ is a multiple of }}n\right\}}\,$

be the set of all positive natural numbers that are multiples of ${}n$. This is a family of subsets of ${}\mathbb {N}$ indexed by the positive natural numbers. We have the inclusions

$\mathbb {N} n\subseteq \mathbb {N} m{\text{ for }}m{\text{ divides }}n.$

The intersection

$\bigcap _{n\in \mathbb {N} _{+}}\mathbb {N} n$

is empty because no positive natural number is a multiple of every positive natural number (${}0$ is such a multiple).