Topological Space is Quasiuniformizable

Theorem
Every topological space is quasiuniformizable.

Proof
Let $T = \struct {S, \tau}$ be a topological space.

Let $\BB$ be defined as:
 * $\BB := \set {u_G: u_G = \paren {G \times G} \cup \paren {\paren {S \setminus G} \times G}, G \in \tau}$

Then $\BB$ is a filter sub-basis for a quasiuniformity on $S$ such that $\struct {\struct {S, \UU}, \tau}$ is a quasiuniform space.