# Decide whether the set is an open subset of $\mathbb{R}$

Okay so I have to determine whether [2,4] and $\mathbb{R}-\mathbb{Q}$ are open subsets of $\mathbb{R}$.

The definition of an open set that I was given is:

A subset U of $\mathbb{R}$ is called an open set if $U=\emptyset$ or if for each $x \in U$ there is an open interval I such that $x\in I \subseteq U$

So with [2,4] I know that it is a closed interval but I'm not sure how to determine if it is an open subset of $\mathbb{R}$.

Could anyone possibly explain why [2,4] and $\mathbb{R}-\mathbb{Q}$ would either be open subsets or not of $\mathbb{R}$

• Hint: try to apply the definition at some special points of [2,4]. – Did Sep 25 '13 at 5:31

• Since $2 \in [2,4]$ any open ball with center $2$ will be of the form $(2-\epsilon,2+\epsilon)$.
• Consider an $\sqrt{2}\in\mathbb{R}\setminus\mathbb{Q}$. An open ball will with center $\sqrt{2}$ will be of the form $(\sqrt{2}-\epsilon,\sqrt{2}+\epsilon)$. Now use the fact that $\mathbb{Q}$ is dense in $\mathbb{R}$.
• Does $[2,4]$ contain any open interval $(a,b)$ such that $2\in(a,b)\subseteq[2,4]$? Why?
• Does $\Bbb R\setminus\Bbb Q$ contain any open interval $(a,b)$ such that $\sqrt2\in(a,b)\subseteq\Bbb R\setminus\Bbb Q$? Why?