Cardinality of Basis of Sorgenfrey Line not greater than Continuum

Theorem
Let $T = \left({\R, \tau}\right)$ be the Sorgenfrey Line.

Let
 * $\mathcal B = \left\{{\left[{x \,.\,.\, y}\right): x, y \in \R \land x < y}\right\}$

be the basis of $T$.

Then $\left\vert{\mathcal B}\right\vert \leq \mathfrak c$

where
 * $\left\vert{\mathcal B}\right\vert$ denotes the cardinality of $\mathcal B$,
 * $\mathfrak c = \left\vert{\R}\right\vert$ denotes continuum.