Past exam of the analytic topology and axiomatic set theory course of the University of Oxford / 2009 / A / 1 / v

If the origin had a countable local basis $U_1,U_2,\dots$, replace it by the decreasing local basis $V_n={\bigcap }_{j\le n}{U}_j$. Each $V_n$ meets the dense set $Y$, so choose $p_n\in V_n\cap Y$. Given any neighbourhood $U$ of the origin, some $U_j$ is contained in $U$; for every $n\ge j$ we then have $p_n\in V_n\subseteq U_j\subseteq U$. Thus $p_n$ converges to the origin, contradicting the preceding part. Consequently first countability fails already at the origin, and $X$ is not first countable.