Open set in terms of nets

Question

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.

Answer 1

Assume that $U$ is not open. Then $X \setminus U$ is not closed and we find $x \in U$ such that $x \in \overline{X \setminus U}$. Choose a net $(x_\alpha)$ in $X \setminus U$ such that $x_\alpha \to x$. By assumption some $x_\alpha \in U$ which is impossible.

Therefore $U$ must be open.