Point in Set Closure iff Limit of Net

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

Let $E \subseteq X$ be a subset.

Let $\map \cl E$ be the closure of $E$ in $\struct {X, \tau}$.

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

Necessary Condition
Let $x \in \map \cl E$.

Let $\Lambda$ be the set of open neighborhoods of $x$ in $\struct {X, \tau}$.

From Open Neighborhoods of Point form Directed Ordering, $\struct {\Lambda, \supseteq}$ is a directed set.

For each $U \in \Lambda$, we have $U \cap E \ne \O$ from the definition of closure.

So for each $U \in \Lambda$ we can pick $x_U \in U \cap E$, forming a Moore-Smith sequence $\family {x_U}_{U \in \Lambda}$ valued in $E$.

We show that $\family {x_U}_{U \in \Lambda}$ converges to $x$.

Let $V$ be an open neighborhood of $x$ in $\struct {X, \tau}$.

Then if $U \supseteq V$, we have $x_V \in V \cap E \subseteq U \cap E \subseteq U$.

Hence $\family {x_U}_{U \in \Lambda}$ converges to $x$.

Sufficient Condition
Let $x \in X$.

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

Let $U$ be an open neighborhood of $x$ in $\struct {X, \tau}$.

Then since $\family {x_\lambda}_{\lambda \in \Lambda}$ converges, there exists $\lambda_0 \in \Lambda$ such that:


 * $x_\lambda \in U$ for all $\lambda \in \Lambda$ with $\lambda_0 \preceq \lambda$.

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

So $U \cap E \ne \O$ for each open neighborhood $U$ of $x$.

So $x \in \map \cl E$ from the definition of closure.