TITLE: It is possible to write open subsets in the form $\Omega=\{x\in \mathbb{R}^N:g(x)>0 \}$
QUESTION [8 upvotes]: Let $\Omega$ be an open subset of $\mathbb{R}^N$. Do we have in general the existence of a continuous function $g:\mathbb{R}^N\to\mathbb{R}$ such that
  $$\Omega=\{x\in \mathbb{R}^N:g(x)>0 \}$$

REPLY [9 votes]: Hint: If $\Omega\neq \mathbb R^N$
 $$g:x\mapsto d(x,\Omega^c)$$
and If $\Omega=\mathbb R^N$
$$g(x)  = \|x\| +1$$