I am familiar with the notation $C^\infty_p(M)$, which denotes the algebra of germs of $C^\infty$ functions at $p$, where two functions defined on a neighborhood of $p$ are equivalent if they agree on some, possible smaller neighborhood of $p$.
On wikipedia here: https://en.wikipedia.org/wiki/Germ_(mathematics)#Examples
it is stated that $C^\infty(M)$ is the space of germs of smooth functions defined on the entirety of $M$. I'm a bit confused by this, because I don't see what the equivalence relation would be. Lee's smooth manifolds makes no such claim, it just says $C^\infty(M)$ is the algebra of smooth functions.
My question is are the elements of $C^\infty(M)$ usually taken to be smooth functions, or germs of smooth functions? And if they are germs, under what equivalence relation?
Edit: I was very likely misreading the wikipedia page.
Best Answer
Short answer: The elements of $C^\infty(M)$ are smooth functions from $M$ to $\mathbb R$; there is no equivalence relation involved.
I think you're misreading the Wikipedia article. It doesn't say "$C^\infty(M)$ is the space of germs of smooth functions defined on the entirety of $M$." What it actually says (with some irrelevant intervening text deleted) is
In the special case that $X$ is a smooth manifold and $Y=\mathbb R$, what this means is that for each open subset $U\subseteq X$, we define $C^\infty(U)$ to mean the set of smooth functions from $U$ to $\mathbb R$, and then for each $x\in M$, we use the equivalence relation described earlier in the article to construct the space $C^\infty_x$ of germs of smooth functions at $x$.