[Math] True or False Questions about Open , Closed, and Closure Set.

Question

1. For any set A, complement of closure A is open => True since closure A is closed and complement must be open.

2. If a set A has an isolated point, it cannot be an open set. => I think it is true, but can not think of proof.

3.Set A is closed if and only if A= closure of A => True by following definition.

4. If A is a bounded set, then s=supA is a limit point of A => False Let A={1,2,3} then supA is 3 but 3 is not a limit point.

5. Every finite set is closed => true, singleton element is closed and finite union of closed sets is closed.

6. An open set that contains every rational number must necessarily be all of R
   I think it is true, but anyone can explain?

If there are any wrong answers, please point out and show me some counterexmaples.

Thanks

Answer 1

1. Looks good.

2. This is true in $\mathbb{R}$, but not in general metric spaces. For a trivial example, consider the discrete metric on a single point; for a slightly less trivial example, consider the set $\{1\}\subset\{1/n:n\in\mathbb{Z}^+\}$ with the metric induced from $\mathbb{R}$.

3. Looks good.

4. Looks good.

5. True in metric spaces but not in topological spaces, for instance by taking the indiscrete topology.

6. False, consider $(-\infty,\sqrt{2})\cup(\sqrt{2},\infty)$. There are stranger examples; for instance, if we enumerate $\mathbb{Q}=(q_i)_{i\in\mathbb{N}}$, then $\bigcup (q_i-2^{-i},q_i+2^{-1})$ is open and contains the rationals, but the sum of the lengths of these intervals is merely $4$!