So I have a topology defined as follows: $$ \tau = \{\mathbb{R} \} \cup \{ U \subset \mathbb{R} \ \ | \ \ 0 \notin U \}$$
I have already prooved that is a topology of $\mathbb{R}$ and that the local basis of $x \in \mathbb{R}$ are defined by:
$\{\mathbb{R}\}$ if $x=0$ and $\{x\}$ if $x \neq 0$.
However I am having troubles trying to prove wether $(\mathbb{R}, \tau)$ is first-countable or not.
Context.- I think that it is first-countable because every point in $\mathbb{R} $ has a countable neighbourhood basis, although I am not sure if this is correct because if $x =0$, is $\{\mathbb{R}\}$ countable?