Characterization of Closedness in terms of Nets

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

Let $E \subseteq X$ be a subset.

Then $E$ is closed :


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

Proof
From Set is Closed iff Equals Topological Closure, we have that $E$ is closed :


 * $x \in E$ $x \in \map \cl E$

From Point is in Topological Closure iff Limit of Moore-Smith Sequence, we have $x \in \map \cl E$ :


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