1. ## open set

Are the following are open sets in . In each case say why?

1.[0,)

2. -

3.

For 1. I know its not an open set because the complement is open so [0,) is closed.

But need help for 2 and 3.

Thanks

2. Originally Posted by 1234567
Are the following are open sets in . In each case say why?

1.[0,)

2. -

3.

For 1. I know its not an open set because the complement is open so [0,) is closed.

But need help for 2 and 3.

Thanks
How are you defining open here? the natural way? open subsets of the reals are open intervals? if so,

yes, 1. is closed, because its compliment $\displaystyle (-\infty, 0)$ is open.

For 2, notice that you have the set:

$\displaystyle \dots (-2, -1) \cup (-1,0) \cup (0,1) \cup (1,2) \cup \dots$

For 3, $\displaystyle \mathbb Q$ is a set of points on the real line... but be careful, $\mathbb Q$ is dense in $\mathbb R$, so you're going to want to come from the perspective of boundary points, etc

3. Yes open subsets of the reals are open intervals.

Thanks for the help.

For 3. I know the complement is the set of all irrational numbers which is not open so do that mean $\displaystyle \mathbb Q$ is also not open.

4. Originally Posted by 1234567
Yes open subsets of the reals are open intervals.
Actually any open set is is the union of open intervals.