Is $\Omega$ not open? Suppose $\Omega\subset\mathbb{R}^d$ is connected.  Let $z:[0,1]\to\Omega$ be a continuous path.  Suppose $\underset{t\in[0,1]}{\inf}\text{dist}(\partial\Omega,z(t))=0$.  Is $\Omega$ necessarily not open?

Remark that the image of $[0,1]$ in $\Omega$ is compact, so the function $t \mapsto \textrm{dist}(\partial \Omega, z(t))$ attains its minimum i.e. the image of $[0,1]$ has nontrivial intersection with the boundary. In other words, at least one element of the boundary is in $\Omega$, and hence $\Omega$ cannot be open. 
