Sorgenfrey Line is First-Countable

Theorem
Let $\R$ be the set of real numbers.

Let $\mathcal B = \left\{ {\left[{a \,.\,.\, b}\right) : a, b \in \R}\right\}$.

Let $\tau$ be the topology generated by $\mathcal B$, that is, the Sorgenfrey line.

Then $\tau$ is first-countable.

Proof
Let $\mathcal B_x = \{ \left[x \,.\,.\, x + \frac{1}{n}\right) | n \in \N_{>0}\}$.

By definition of first-countability, it suffices to show that:


 * (1): $\mathcal B_x$ is countable
 * (2): $\mathcal B_x$ is a local basis at $x$

(1) follows from the fact that $\mathcal B_x$ is a bijection from the set of natural numbers.

(2):

By definition of local basis, it suffices to show that $\forall U \in \tau: x \in U: \exists B \in \mathcal B_x: B \subseteq U$.

Pick any $U$ in $\tau$.

By definition of $\tau$, there exists $[x \,.\,.\, x + \varepsilon)\subseteq U$ for some $\varepsilon \in \R_{>0}$.

By Archimedean Principle there exists $n\in \N$ such that $n > \dfrac 1 {\varepsilon}$ (i.e. $\dfrac 1 n < \varepsilon$).

So $x \in \left[x \,.\,.\, x + \dfrac 1 n\right) \subseteq \left[x \,.\,.\, x + \varepsilon\right) \subseteq U$.

Also see

 * Sorgenfrey Line is not Second-Countable
 * Sorgenfrey Line is Separable