0
$\begingroup$

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 $\mathbb{R}$

$\endgroup$
  • $\begingroup$ Hint: try to apply the definition at some special points of [2,4]. $\endgroup$ – Did Sep 25 '13 at 5:31
1
$\begingroup$

Hints:

  • 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 $\mathbb{R}$.

$\endgroup$
0
$\begingroup$

HINT:

  • 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?

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.