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 $\BB$ is a basis for $\tau$.

By definition of basis:

$\forall U \in \tau: \forall x \in U: \exists B \in \BB: x \in B \subseteq U$

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

Now:

$\forall x \in \R: \exists B_x \in \BB: x \in B_x \subseteq \hointr x {x + \epsilon}$

This $\BB_x$ has an infimum equal to $x$.

So for different $x$, the corresponding $\BB_x$ is different.

So the cardinality of $\BB$ is at least $\size \R$, which is uncountable.

$\blacksquare$

Also see

• Sorgenfrey Line is First-Countable
• Sorgenfrey Line is Separable