Let $V$ be a locally closed subset of $\mathbb{R}^n$, that means the intersection of a Zariski closed subset of $\mathbb{R}^n$ with a Zariski open subset of $\mathbb{R}^n$. My question is how one can extend a smooth function $f:V\longrightarrow \mathbb{R}$ to the whole space, since the domain is in general not closed both in euclidean and in the Zariski topology.
I'm studying on "Topology of Real Algebraic Sets" by S.Akbulut and H.King and at page 58 they use naturally this result that is not so trivial by my knowledge about smooth functions on locally closed sets.
I really thank everyone that would give me some hints or references where I may find an answer to my question.