A and B are subsets of the reals; neither A nor B is an open set, but ∪ is an open set.

Question

Either give an example that satisfies the statement or explain why no such example is possible. The context is the reals.

A and B are subsets of the reals; neither A nor B is an open set, but ∪ is an open set.

So for this one If you have 2 non open sets, I feel like then I would assume these are both closed sets. I don't think 2 closed sets can form an open set. I just don't know how to go about proving it.

Answer 1

1. It is not necessarily the case that all non-open sets are closed. Take half-open intervals such as $(a,b]$ for example. That might help motivate an example.
2. Recall that $\Omega\subseteq\mathbb{R}$ is closed if $\Omega^c=\mathbb{R}\setminus\Omega$ is open. What must the complement of a finite subset of $\mathbb{R}$ be?
3. Remember that for $\Omega\subseteq\mathbb{R}$ to be open, $\text{Int}(\Omega)=\Omega$. What does that tell you about the non-empty intersection of two open sets (that also must be open)?
4. Your intuition is good! Closed sets including their boundary is what lets this happen.

I hope this pushes you in the right direction!