Quotient topology, product topology and subspace topology

Question

Let $S$ be the set $\mathbb R^2\setminus\{(0, y) \mid y \neq 0\}$.

Let $\tau_1$ denote the subspace topology on $S$ induced from the usual topology of $\mathbb{R}^2$.

Now, consider the surjective map $p : \mathbb{R}^2 \to S$ defined by $(x, y) \mapsto (x, y)$ for $x \neq 0$, and $(0, y) \mapsto (0, 0) $, $\forall y$.

Let $\tau_2$ be the resulting quotient topology on $S$. (In other words, $\tau_2$ is the finest topology on $S$ with respect to which $p$ is continuous.)

1. Is $\tau_2$ metrizable?

2. Is $\tau_2$ 2nd countable?