I am reading a proof and it says to verify the following: Suppose $Z$ is a separable Banach space and $F$ is a closed subset of $Z$. Let $\mathcal{O}$ be a countable basis of open subsets of $Z$. We have $x,y\in F \implies x+y\in F$ iff for all $O,O'\in\mathcal{O}$ with $O\cap F\neq \emptyset$ and $O'\cap F\neq \emptyset$, we have $(O+O')\cap F \neq \emptyset$.
It is clear that => holds. I am having trouble with the other direction. Any help would be appreciated.
Suppose that $x,y\in F$, but $x+y\notin F$. $F$ is closed, so $x+y$ has an open nbhd $U$ such that $U\cap F=\varnothing$. Addition is a continuous function from $Z\times Z$ to $Z$, so there are $O,O'\in\mathcal{O}$ such that $x\in O$, $y\in O'$, and $O+O'\subseteq U\subseteq Z\setminus F$.