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

Question

I am trying to understand this proof from Munkres' book which shows that the Sorgenfrey line does not have a countable basis. His proof is:

Let $\beta$ be a basis for $\mathbb{R}_l$. Choose for each $x$ an element $B_x \in \beta$ such that $x \in B_x \subset [x, x+1)$. If $y$ is not equal to $x$ then $B_y \neq B_x$ since inf($B_x$) = $x$ and $y$ = inf($B_y$). Therefore $\beta$ must be uncountable.

I am new to topology so I am probably just missing a simple point but I do not see why $\beta$ must be uncountable according to this argument. I agree that $B_y \neq B_x$ but isn't it possible that $y \in B_x$ (and thus the set $B_y$ is not needed as part of the basis)? Thanks in advance for your help.

Answer 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 $|\beta|\ge |\Bbb R|=\mathcal c$.