Closure of a Set is all its Net Limits
# Statement
Let $X$ be a Topological Space and let $S \subset X$. Then $$\text{cl}S = {x \in X : x_{\alpha} \to x \text{ for some net } (x_{\alpha})_{\alpha \in A} \subset X}$$
# Proof
Closure of a Set is all its Limit Points and A point is a Limit Point iff it is a Net Limit. $\square$