I won’t be posting the problems themselves because these problems aren’t mine own but from elsewhere. Links or source will be posted so you can look up the problems there.

Solutions below for Munkres’ Topology $$\S 13$$

1. If for every $$x\in A$$, there is an open set $$x\in U, U \subset A$$. Then let, $$\mathcal{U}$$ be the set of all such $$U’s$$. Then $$\mathcal{U}$$ is at most uncountable. But the union of an uncountable number of open sets is still open, so $$\cup U_i$$ is still open. Furthermore, since every $$U_i\subset A$$, then $$\cup U_i = A$$. So, that $$A$$ is an open set.