If $A\subset\mathbb{R^2}$ is countable, is $\mathbb{R^2}\setminus A$ path connected? [duplicate]
Here is a marginal variation:
Choose $x_0,x_1 \in A^c$. Let $d \neq 0$ be orthogonal to $x_1-x_0$. Define the collection of paths $\gamma_\alpha(t) = x_0 + t(x_1-x_0) + \alpha t(1-t) d$. Since $\{\gamma_\alpha\}_{\alpha \in [0,1]}$ is uncountable, there exists $\alpha_0 \in [0,1]$ such that $\gamma_{\alpha_0}[0,1] \cap A = \emptyset$. Hence $A^c$ is path connected.