Weight of Sorgenfrey Line is Continuum

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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:

$\mathfrak c \le \map w T$

Aiming for a contradiction, suppose

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

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:

$\hointr x y \notin \tau$

$\blacksquare$


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