Suppose we have an (affine) irreducible algebraic variety $X$ over $\mathbb{R}$, and we are given one polynomial $f$. Consider the closed subset $C$ where this is zero. What can we say about the number of connected components of $X \setminus C$ with respect to the Euclidean topology? What are essential data about $f$ to look at?
I have heard about a proof of the Jordan-curve theorem that $\mathbb{R}^2 \setminus Y$, with $Y \sim S^1$ via a differentiable map, has exactly two connected components, which use cohomological tools; any insight about a connection between my question and cohomology of something?
Thank you, Andrea