Possible typographical error in Michael Searcoid's book?

Question

Let $\overline{S}$ denote closure of $S$ and $S^\circ$ denote the interior of $S$. The theorem did not state that we are dealing with subsets $U$ in $S$, but the proof makes this assumption. If it's not a typo then I would like to know why we can make this assumption.

Answer 1

As Théophile comments, it's an assumption for the sake of argument, not an assumption.

What you want to prove is that $S^\circ$ is the largest open subset of $S.$ This means that any open subset of $S$ is contained in $S^\circ.$ You start a proof of this by saying "let $U$ be an (arbitrary) open subset of $S$..." and then show that $U\subseteq S^\circ.$