I have 2 questions hopefully someone can help with.

1) Is R the only open set containing Q? Prove or give an example of a set containing Q.

I do believe R is the only open set containing Q but I have no idea how to prove this.

2) Is the set of integers a closed set? I'm pretty sure it is, since the boundary points are the integers. But is is also open?

Thanks so much for the help!