I am trying to prove that the following topology in $\mathbb{R}$ is not regular. It is defined as follows:

For $x\neq0$, a neighborhood base for $x$ is determined by $(x-\alpha,x+\alpha)$ with $\alpha>0$.

For $x=0$, a neighborhood base is $B\_\alpha=\{t\in \mathbb{R}|-\alpha<t<\alpha,t\neq1,\frac{1}{2},\frac{1}{3}...\}$.

I have thought of taking the closed set $C=\{1,\frac{1}{2},\frac{1}{3}...\}$ as a counterxample of the definition of the regular space but I don't see how to showitformally. Appreciate some help. Thanks.