Which is the definition of the set of germs $C_p^{\infty}(\mathbb R^n)$? Does $C^{\infty}(U)$ consist of germs or functions?

Question

Which one is the set known as $C_p^{\infty}(\mathbb R^n)$?

1. The set of germs of smooth real-valued functions defined on $\mathbb R^n$

2. The set of germs of smooth real-valued functions defined on a fixed open subset of $\mathbb R^n$ that contains $p$

3. The set of germs of smooth real-valued functions defined on any open subsets of $\mathbb R^n$ that contains $p$

My book sounds like it's saying (1) and then later (3). Is the language of the book actually identifying $C_p^{\infty}(\mathbb R^n)$ as (1) throughout?

It says

We write $C_p^{\infty}(\mathbb R^n)$, or simply $C_p^{\infty}$ if there is no possibility of confusion, for the set of all germs of $C^{\infty}$ functions on $\mathbb R^n$ at $p$. $\tag{7}$

I think this should be

We write $C_p^{\infty}(\mathbb R^n)$, or simply $C_p^{\infty}$ if there is no possibility of confusion, for the set of all germs of $C^{\infty}$ functions on open subsets of $\mathbb R^n$ that contain $p$.

Later, my book talks about $C^{\infty}(U)$ for an open subset $U$ of $\mathbb R^n$. Which one is $C^{\infty}(U)$?

4. The set of germs of smooth real-valued functions defined on $U$

5. The set of germs of smooth real-valued functions defined on a fixed open subset of $U$ (which in turn is an open subset of $\mathbb R^n$)

6. The set of germs of smooth real-valued functions defined on any open subsets of $U$ (which in turn is an open subset of $\mathbb R^n$), thus functions from different germs may have disjoint domains.

My book says

The ring of $C^{\infty}$ functions on an open set $U$ is commonly denoted by $C^{\infty}(U)$

If $C^{\infty}(U)$ is (4) instead of (6), then I think $C_p^{\infty}(\mathbb R^n)$ should be (1) instead of (3) because of the language in $(7)$.

• Edit: Actually, does $C^{\infty}(U)$ still consist of germs rather than functions?

My book is An Introduction to Manifolds by Loring W. Tu.

Answer 1

First, $C^\infty(U)$ is the set of all smooth functions defined on $U$, not the set of germs at $p$. The set of germs at $p \in U$ is denoted by $C_p^\infty(U)$.

For the rest, you are correct that the author is being a little lax in his terminology, but the reason he isn't bothering to be exact is that it makes no difference. One gets an equivalent set of germs with all three definitions.

That is, if we have a point $p$, and a fixed open set $U_p$ with $p\in U_p \subseteq \Bbb R^n$, and define the three equivalences as in the definition of germs for the three collections

• $\sim_1$ on $C^\infty(\Bbb R^n)$, with $C_1 = C^\infty(\Bbb R^n)/ \sim_1$
• $\sim_2$ on $C^\infty(U_p)$, with $C_2 = C^\infty(U_p) / \sim_2$
• $\sim_3$ on $\mathscr F = \{f \mid V \in \mathscr O(\Bbb R^n) \wedge p \in V \wedge f\in C^\infty(V)\}$, with $C_3 =\mathscr F / \sim_3$

Then there is a natural one-to-one correspondence between the three sets of germs. Any $f \in C^\infty(\Bbb R^n)$ also is a member of $\mathscr F$, and the restriction $f|_{U_p}$ is in $C^\infty(U_p)$. And if $g$ is another such function, then it is obvious that $$f \sim_1 g \iff f|_{U_p} \sim_2 g|_{U_p} \iff f \sim_3 g$$ which induces injections of $C_1$ into $C_2$ and $C_3$. Similarly, $C^\infty(U_p) \subseteq \mathscr F$, which also induces an injection of $C_2$ into $C_3$. However, for any $f \in \mathscr F$, it is not hard to show that for some $g \sim_3 f, g$ is the restriction of some $g' \in C^\infty(\Bbb R^n)$. This induces an injection of $C_3$ into $C_1$, which is the inverse of the injection of $C_1 \to C_3$.

Since the elements of $C_1, C_2, C_3$ are all naturally identifiable with each other, we can consider any of the sets to be the set of germs at $p$.

Now by careful reading of the text, it is apparent that Prof. Tu is actually defining the set of germs $C_p^\infty(\Bbb R^n)$ to be $C_3$, the one you describe. But because he knows it doesn't matter which is used, he got a little careless in his wording when he actually introduces the set.