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.

  • $\begingroup$ 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). $\endgroup$ – Dog_69 Feb 7 at 19:45
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    $\begingroup$ @Dog_69 I don't understand your comment. The condition is required to hold for all nets, not just one in particular. $\endgroup$ – geodude Feb 7 at 20:15
  • $\begingroup$ Yes I misunderstood the question. Sorry. $\endgroup$ – 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.


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