General Topology - First and Second Countable Consider the topology $\tau(\mathcal{E})$ on $\mathbb{R}$ generated by $\mathcal{E} = \{N(x,\epsilon) : x \in \mathbb{R}, \epsilon > 0\}$ and 
$
N(x;\epsilon) = \left\{
\begin{array}{ll}
      \mathbb{Q} \cap (x - \epsilon, x + \epsilon) & \text{ if } x \in \mathbb{Q} \\
      (x - \epsilon, x + \epsilon) \setminus \mathbb{Q} & \text{ if } x \in \mathbb{R} \setminus \mathbb{Q} \\
\end{array} 
\right.$
Is this space first and second countable? 
Here are my ideas:
I think it is not first countable because for all $x \in \mathbb{Q}$ and $x \in \mathbb{P}$, $N(x,\epsilon)$ can act as a nbd base at $x$. However, $\forall x \in \mathbb{P}$, the nbd base $N(x,\epsilon)$ is uncountable. 
Then if the space is not first countable then it is not second countable.
 A: It's second countable. Let $x_n$ be an injective sequence that ranges over $\Bbb{Q}$, and $y_n$ be an injective sequence that is dense in $\Bbb{R} \setminus \Bbb{Q}$. Form a basis for the topology like so:
$$B = \left\{\Bbb{Q} \cap \left(x_n - \frac{1}{m}, x_n + \frac{1}{m}\right) : n, m \in \Bbb{N}\right\} \cup \left\{\left(y_n - \frac{1}{m}, y_n + \frac{1}{m}\right) \setminus \Bbb{Q} : n, m \in \Bbb{N}\right\}.$$
Note that $B$ is countable. It's clear that every neighbourhood $\Bbb{Q} \cap \left(x - \varepsilon, x + \varepsilon\right)$, where $x \in \Bbb{Q}$, contains a set in $B$, as $x = x_n$ for some $n$.
On the other hand, consider some $y \in \Bbb{R} \setminus \Bbb{Q}$, and $\varepsilon > 0$. Since $y_n$ is dense in $\Bbb{R} \setminus \Bbb{Q}$, there must be some $n$ such that $|y - y_n| < \frac{\varepsilon}{2}$. Choose $m$ such that $\frac{1}{m} < \frac{\varepsilon}{2}$ Then,
$$\left(y_n - \frac{1}{m}, y_n + \frac{1}{m}\right) \setminus \Bbb{Q} \subseteq \left(y_n - \frac{\varepsilon}{2}, y_n + \frac{\varepsilon}{2}\right) \setminus \Bbb{Q} \subseteq \left(y - \varepsilon, y + \varepsilon\right) \setminus \Bbb{Q}.$$
Thus, every neighbourhood in the original basis is contains a set from $B$, thus $B$ is also a basis, and countable to boot.
