Let $X$ be a topological space, and $U\subseteq X$ be a subset of $X$ with the following property:
For every convergent net $x_\alpha\to x$ in $X$ such that $x\in U$, there exists an $\alpha$ such that $x_\alpha\in U$.
Is then $U$ open?
Note that I am not requiring that $x_\beta\in U$ for $\beta\ge\alpha$.
If this is false, what is a counterexample? I feel like I'm missing something obvious.