Sorgenfrey Line is not Second-Countable

Theorem
Let $T = (\mathbb R, \tau)$ 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 $x \in \mathbb R$, pick $U = [x, x+\varepsilon) \in \tau$ for some $\varepsilon > 0$.

Now $\forall x \in \mathbb R: \exists B_x \in \mathcal B: x \in B_x \subseteq [x, x+\varepsilon)$

This $\mathcal B_x$ has infimum equals to $x$.

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

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

Also see

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