Cardinality of Basis of Sorgenfrey Line not greater than Continuum

Theorem

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

Let

$\BB = \set {\hointr x y: x, y \in \R \land x < y}$

be the basis of $T$.

Then $\card \BB \le \mathfrak c$

where

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

$\mathfrak c = \card \R$ denotes the continuum.

Define a mapping $f: \BB \to \R \times \R$:

$\forall I \in \BB: \map f I = \tuple {\min I, \sup I}$

That is:

$\map f {\hointr x y} = \tuple {x, y} \forall x, y \in \R: x < y$

We will show that $f$ is an injection by definition.

Let $I_1, I_2 \in \BB$ such that:

$\map f {I_1} = \map f {I_2}$

$\ds I_1$

$=$

$\ds \hointr {\min I_1} {\sup I_1}$

$\ds $

$=$

$\ds \hointr {\min I_2} {\sup I_2}$

by $\map f {I_1} = \map f {I_2}$

$\ds $

$=$

$\ds I_2$

So:

$I_1 = I_2$

Thus $f$ is an injection.

$\card \BB \le \card {\R \times \R}$

$\card {\R \times \R} = \map \max {\mathfrak c, \mathfrak c}$

Thus:

$\card \BB \le \mathfrak c$

$\blacksquare$