Let denote by $U_n\subset \mathbb R^2$ a sequence of open bounded neigborhoods of $X$, so that 
$$U_{n+1}\subset U_n\ \ \text{and}\ \  \bigcap_n U_n=X.$$
We can assume that all $U_n$ are connceted and therefore path-connected. Coose a point $p\in X$ distict from $x$ and consider a sequence of paths $\gamma_n$ in $U_n$ from $p$ to $x$. Fix $\epsilon>0$ such that $\epsilon<|p-x|$.
For each path choose the smalest value $t_n\in[0,1]$ so that $|\gamma_n(t_n)-x|=\epsilon$.
The image $Z_n=\gamma([0,t_n])$ is connected compact set.
Let $Z$ be a Hausdorff limit of a subsequence of $Z_n$.
Note that $Z$ is a compact connected subset of $X$.
Clearly, $Z\not\ni x$ and it contains at least two points; a contradiction