A naive question inspired by the idea in algebraic geometry.
Let $M$ be a smooth manifold (second countable and Hausdorff), not assuming compactness. Let $C^\infty(M)$ be the ring of smooth functions. Define an $\mathbb R$-point of $M$ as a ring map $$C^\infty(M)\to\mathbb R. $$
The first examples of $\mathbb R$-points are the real points, i.e. for $p\in M$, the evaluation is an $\mathbb R$-point $$\mathrm{ev}_p:C^\infty(M)\to\mathbb R,\quad f\mapsto f(p). $$
The existence of local cut-off function and coordinate functions proves that $p\mapsto \mathrm{ev}_p$ is injective.
I want to understand the relations between $\mathbb R$-points and real points of $M$. Here are some natural questions:
- Is $p\mapsto \mathrm{ev}_p$ surjective? I tend to believe it false if $M$ is not compact. But counterexamples are not available to me.
- If the truth above depends on $M$, can you give sufficient conditions (or examples) in both cases?
- Any meaningful geometric structure on the $\mathbb R$-points, such as a topology?
Thanks.
