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 $\displaystyle \displaystyle (-\infty, 0)$ is open.
For 2, notice that you have the set:
$\displaystyle \displaystyle \dots (-2, -1) \cup (-1,0) \cup (0,1) \cup (1,2) \cup \dots$
For 3, $\displaystyle \displaystyle \mathbb Q$ is a set of points on the real line... but be careful, $\displaystyle \mathbb Q$ is dense in $\displaystyle \mathbb R$, so you're going to want to come from the perspective of boundary points, etc