# Open set in terms of nets

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.

• There are many of them. A constant net on a $T_1$ space, where every singleton closed. The set of acumulation points of a net in a connected space (to ensure that it isn't clopen). – Dog_69 Feb 7 at 19:45
• @Dog_69 I don't understand your comment. The condition is required to hold for all nets, not just one in particular. – geodude Feb 7 at 20:15
• Yes I misunderstood the question. Sorry. – Dog_69 Feb 7 at 21:09

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.