Indexed set family/Multiples of n/Example

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

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

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

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

The intersection

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

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