Homology of the complement of a real hypersurface

Question

Consider a real algebraic set $Z(f) = \{x \in \Bbb{R}^n\,|\, f(x) = 0\} \subset \Bbb{R}^n$ (not necessarily irreducible). I'm thinking about wether the (Euclidean) closure of a connected component of the complement $\Bbb{R}^n\setminus Z(f)$ has vanishing $n$:th homology or not. It feels like this should not be a hard question, but I keep finding myself thinking about an intuitive picture in $n=2$ or $n=3$ (where I think it is vanishing), unable to find a more strict approach.