# Topological Property of the Set of Integers

Is the set of integers $\mathbb{Z}$ closed in $\mathbb{R}$ equipped with the usual topology?

• It depends. What's the topology on $\mathbb{R}$? – user61527 Mar 17 '14 at 20:54
• @T.Bongers My bad, I should've said that I have the usual topology on $\mathbb{R}$. – Rachel Mar 17 '14 at 21:05

$\mathbb{R}\setminus\mathbb{Z} = \bigcup_{n\in\mathbb{Z}}(n,n+1)$ is a union of open sets and therefore open. Since the complement of $\mathbb{Z}$ is open in $\mathbb{R}$, $\mathbb{Z}$ is closed in $\mathbb{R}$.
• This is only correct assuming the usual topology on $\mathbb{R}$. – user61527 Mar 17 '14 at 20:58