Show that the Sorgenfrey line does not have a countable basis.

general-topology second-countable sorgenfrey-line

Solution 1:

Hints:

If you agree $B_x\not= B_y$ when $x\not=y$, then I will ask you how many numbers in $\Bbb R$?

Added:

In fact, there exists a function $f: \Bbb R \to \beta$ such that for any $x\in \Bbb R$, there aways exists $B_x\in \beta$, s.t. $f(x)=\beta_x$. So $|\beta|\ge |\Bbb R|=\mathcal c$.

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