Sorgenfrey Line is not Second-Countable

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

Then $T$ is not second-countable.

Proof
Suppose $\mathcal B$ is a basis for $\tau$.

By definition of basis:
 * $\forall U \in \tau: \forall x \in U: \exists B \in \mathcal B: x \in B \subseteq U$

For all $x \in \R$, pick $U = \hointr x {x + \epsilon} \in \tau$ for some $\epsilon > 0$.

Now:
 * $\forall x \in \R: \exists B_x \in \mathcal B: x \in B_x \subseteq \hointr x {x + \epsilon}$

This $\mathcal B_x$ has an infimum equal to $x$.

So for different $x$, the corresponding $\mathcal B_x$ is different.

So the Cardinality of $\mathcal B$ is at least $\size \R$, which is uncountable.

Also see

 * Sorgenfrey Line is First-Countable
 * Sorgenfrey Line is Separable