Topological Space is Quasiuniformizable

Theorem
Every topological space is quasiuniformizable.

Proof
Let $T = \left({S, \tau}\right)$ be a topological space.

Let $\mathcal B$ be defined as:
 * $\mathcal B := \left\{{u_G: u_G = \left\{{G \times G}\right) \cup \left({\left\{{S \setminus G}\right) \times G}\right), G \in \tau}\right\}$

Then $\mathcal B$ is a filter sub-basis for a quasiuniformity on $S$ such that $\left({\left({S, \mathcal U}\right), \tau}\right)$ is a quasiuniform space.

{{wtd|Prove that above assertion.}}