Answer to Question #348215 in Real Analysis for khoo

Question

prove or disprove that the complement of integers z in R is an open set

Answer 1

For each integer $n$ , open interval $(n,n+1)$ is open and $\mathbb{R}−\mathbb{Z}=⋃_{n∈\mathbb{Z}}(n,n+1)$. That is, $\mathbb{R}−\mathbb{Z}$ is a union of countable many open sets so $\mathbb{R}−\mathbb{Z}$ is open.