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 \( \S 13\)

- 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.