Real Analysis: Determine Whether the Following Sets are Open in $E^1$, closed in $E^1$ or neither.

Question

I just need help getting started so I think I only need help with one of these.

I'm given,

{$x \in \mathbb{R} \ | \ x<0 \ or \ x>1 $}

I know I need to find some open ball (or more simply an open subset) s.t. every element of this subset is contained in the set ($-\infty,0$) and (1,$\infty$)

If I prove that both of these sets are open then I know that the union of the sets must be an open set.

So I create an open ball ($p_0 - r , p_0+r$) $\in$ ($-\infty,0$)

Then I let some $p \in (p_0 - r , p_0+r)$

I want to show that $p \in (-\infty,0)$

Is it enough to show then that p is strictly less than 0?

Thank you in advance.

Answer 1

Let $f:x\mapsto x (x-1) $ from $\Bbb R \to \Bbb R.$.

$f $ is continuous at $\Bbb R$, so the preimage of the open $(0,+\infty)$, which is the set $\{x\in \Bbb R \;\;:\; x <0 \lor x>1\} $ is an open.

Answer 2

What you need to show is that for any element of the set you can find an open ball around it that is in the set. As the union of open sets is open you can concentrate on $(-\infty,0)$ as the proof is similar for the other part. Now assume you are given a point $x \in (-\infty,0)$. Can you find a ball around $x$ that is in the set? You are allowed to have the size of the ball depend on $x$.