Kelley's Theorem

Theorem
Let $\left({D,\preceq}\right)$ be a directed set,

let $S$ be a non-empty set, and

let $n \colon D \to S$ be a net in $S$.

Then $n$ has a universal subnet.

Note
Kelley's Theorem is equivalent to the Axiom of Choice, and leads easily to Tychonoff's Theorem.