# Equivalence of definitions for ring of germs $C_p^{\infty}(\mathbb R^n)$

I want to show the characterizations of $$C_p^{\infty}(\mathbb R^n)$$ are equivalent:

$$A= \{[f] | \text{smooth} \ f: \mathbb R^n \to \mathbb R \}$$

$$B= \{[f] | \text{smooth} \ f: U_p \to \mathbb R \}$$

$$C= \{[f] | \text{smooth} \ f: V_{f,p} \to \mathbb R \}$$

where

• all the germs in $$B$$ can be represented by functions whose domains are a fixed open subset $$U_p$$ of $$\mathbb R^n$$ that contains $$p$$

• while the domaina of functions that represent germs in $$C$$ are any open subsets $$V_{f,p}$$ of $$\mathbb R^n$$ that contain $$p$$.

Note: I am attempting prove $$A=B=C$$ really as equal sets are not merely that they are in bijection.

So far I have done:

$$(A \subseteq B):$$

• Let $$[f] \in A$$ for a smooth $$f: \mathbb R^n \to \mathbb R$$. We must show $$[f] \in B$$, which is done by finding some smooth $$g: U_p \to \mathbb R$$ such that $$[f]=[g]$$, which means $$f|_{W_p} = g|_{W_p}$$ for an a open subset $$W_p \subseteq \mathbb R^n \cap U_p = U_p$$ of $$\mathbb R^n$$ that contains $$p$$.

• We can choose $$g = f|_{U_p}$$ and $$W_p = U_p$$.

$$(B \subseteq C):$$

• Let $$[f] \in B$$ for a smooth $$f: \mathbb U_p \to \mathbb R$$. We must show $$[f] \in C$$, which is done by finding some smooth $$g: V_{p,g} \to \mathbb R$$ such that $$[f]=[g]$$, which means $$f|_{W_p} = g|_{W_p}$$ for an a open subset $$W_p \subseteq V_{p,g} \cap U_p$$ of $$\mathbb R^n$$ that contains $$p$$.

• We can choose $$V_{p,g} = U_p$$ and $$g = f$$.

$$(C \subseteq A):$$

• According to a proposition in Section 13 (11 sections later), an arbitrary smooth extension to the whole manifold, even in the case of $$\mathbb R^n$$ for $$n=1$$, doesn't always exist. However, it exists to say $$C \subseteq A$$:

• Let $$[f] \in C$$ for a smooth $$f: \mathbb V_{p,f} \to \mathbb R$$. We must show $$[f] \in A$$, which is done by finding some smooth $$g: \mathbb R^n \to \mathbb R$$ such that $$[f]=[g]$$, which means $$f|_{W_p} = g|_{W_p}$$ for an a open subset $$W_p \subseteq V_{p,f} \cap \mathbb R^n = V_{p,f}$$ of $$\mathbb R^n$$ that contains $$p$$.

• We can choose $$g = \rho f \ 1_{U_p}$$, where $$\rho: \mathbb R^n \to \mathbb R$$ is a smooth bump function supported in $$U_p$$ and $$\rho(D_p)=\{1\}$$ for some open subset $$D_p$$ of $$\mathbb R^n$$ containing $$p$$, and we can choose $$W_p=D_p$$.

Questions

1. Is anything wrong?

2. Can $$C \subseteq A$$ be proven without bump functions?