Weight of Sorgenfrey Line is Continuum

Theorem
Let $T = \struct {\R, \tau}$ be the Sorgenfrey line.

Then $\map w T = \mathfrak c$

where
 * $\map w T$ denotes the weight of $T$
 * $\mathfrak c$ denotes continuum, the cardinality of real numbers.

Proof
By definition of Sorgenfrey line, the set:
 * $\BB = \set {\hointr x y: x, y \in \R \land x < y}$

is a basis of $T$.

By definition of weight:
 * $\map w T \le \card \BB$

where $\card \BB$ denotes the cardinality of $\BB$.

By Cardinality of Basis of Sorgenfrey Line not greater than Continuum:
 * $\card \BB \le \mathfrak c$

Thus
 * $\map w T \le \mathfrak c$

It remains to show that:
 * $\mathfrak c \le \map w T$


 * $\mathfrak c \not \le \map w T$

Then:
 * $\map w T < \mathfrak c$

By definition of weight, there exists a basis $\BB_0$ of $T$:
 * $\map w T = \card {\BB_0}$

Then by Set of Subset of Reals with Cardinality less than Continuum has not Interval in Union Closure:
 * $\exists x, y \in \R: x < y \land \hointr x y \notin \set {\bigcup A: A \subseteq \BB_0} = \tau$

By definition of $\BB$:
 * $\hointr x y \in \BB \subseteq \tau$

Thus this contradicts by definition of subset with:
 * $\hointr x y \notin \tau$