# Topology question

• Feb 6th 2009, 09:52 PM
horowitz
Topology question
Prove that Cl(Q)=R in the standard topology on R.
• Feb 6th 2009, 10:00 PM
ThePerfectHacker
Quote:

Originally Posted by horowitz
Prove that Cl(Q)=R in the standard topology on R.

Let $a\in R$ for any any $\epsilon > 0$ we know that $(a-\epsilon,a+\epsilon)$ contains both rational and irrational points. Therefore, all points are boundary points. Thus, $\partial \mathbb{Q} = \mathbb{R}$ which means $\text{cl}( \mathbb{Q}) = \mathbb{Q} \cup \partial \mathbb{Q} = \mathbb{R}$.