Characterization of Openness in terms of Nets

Theorem
Let $\struct {X, \tau}$ be a topological space.

Let $U \subseteq X$.

Then $U \in \tau$ for each:
 * $x \in U$
 * directed set $\tuple {\Lambda, \preceq}$
 * Moore-Smith sequence $\family {x_\lambda}_{\lambda \in \Lambda}$ converging to $x$

there exists $\lambda \in \Lambda$ with $x_\lambda \in U$.

Necessary Condition
Suppose that $U \in \tau$.

Take $x \in U$, a directed set $\tuple {\Lambda, \preceq}$ and a Moore-Smith sequence $\family {x_\lambda}_{\lambda \in \Lambda}$ converging to $x$.

By the definition of convergence for Moore-Smith sequences, we have:
 * there exists $\lambda_0 \in \Lambda$ such that for all $\lambda \in \Lambda$ with $\lambda_0 \preceq \lambda$ we have $x_\lambda \in U$.

In particular, $x_{\lambda_0} \in U$.

Sufficient Condition
Suppose that for each:
 * $x \in U$
 * directed set $\tuple {\Lambda, \preceq}$
 * Moore-Smith sequence $\family {x_\lambda}_{\lambda \in \Lambda}$ converging to $x$

there exists $\lambda \in \Lambda$ with $x_\lambda \in U$.

AimForCont $U \not \in \tau$.

Then $X \setminus U$ is not closed.

Note that if we had:


 * if $x \in X$ and for every directed set $\struct {\Lambda, \preceq}$ and Moore-Smith sequence $\family {x_\lambda}_{\lambda \in \Lambda}$ in $X \setminus U$ converging to $x$, then $x \in X \setminus U$.

we would have:


 * $x \in X \setminus U$ $x \in \map \cl {X \setminus U}$

from Point is in Topological Closure iff Limit of Moore-Smith Sequence.

We would then have:


 * $X \setminus U = \map \cl {X \setminus U}$

From Set is Closed iff Equals Topological Closure, we have $\map \cl {X \setminus U} \ne X \setminus U$.

So:


 * there exists $x \in U$, a directed set $\struct {\Lambda, \preceq}$ and a Moore-Smith sequence $\family {x_\lambda}_{\lambda \in \Lambda}$ in $X \setminus U$ converging to $x$.

But from hypothesis, we have $x_\lambda \in U$ for some $\lambda \in \Lambda$.

This is a contradiction.

So we have $U \in \tau$.