The weight of Sorgenfrey line

Question

What is the weight of Sorgenfrey line $S$?

Weight $(X)=\min\{|\mathcal{B}|: \text{ $\mathcal{B}$ a base for $X$}$} + $\omega$

Thanks ahead.

Answer 1

The weight of the Sorgenfrey line is $\mathfrak{c} = 2^{\aleph_0}$. To see this, note that a base of minimal cardinality can be found as a subset of the standard basis $$\mathcal{B} = \{ [ a , b ) : a < b \in \mathbb{R} \}.$$ If you have a subset $\mathcal{B}_0 \subseteq \mathcal{B}$ of cardinality $< \mathfrak{c}$, then there must be a real number $a$ which is not the left-endpoint of any interval in $\mathcal{B}_0$, and it can be shown that, for example, the open set $[ a , a+1 )$ is not a union of sets of $\mathcal{B}_0$, and so $\mathcal{B}_0$ is not a base for the Sorgenfrey line.