TITLE: Sanity check about Wikipedia definition of differentiable manifold as a locally ringed space
QUESTION [37 upvotes]: Most textbooks introduce differentiable manifolds via atlases and charts. This has the advantage of being concrete, but the disadvantage that the local coordinates are usually completely irrelevant- the choice of atlas and chart is arbitrary, and rarely if ever seems to play any role in differential geometry/topology.
There is a much better definition of differentiable manifolds, which I don't know a good textbook reference for, via sheaves of local rings. This definition does not involve any strange arbitrary choices, and is coordinate free. Paragraph 3 in Wikipedia (which is the actual definition) states:

A differentiable manifold (of class $C_k$) consists of a pair $(M, \mathcal{O}_M)$ where $M$ is a topological space, and $\mathcal{O}_M$ is a sheaf of local $R$-algebras defined on $M$, such that the locally ringed space $(M,\mathcal{O}_M)$ is locally isomorphic to $(\mathbb{R}^n, \mathcal{O})$.

This confuses me, because I don't see why such a sheaf should be acyclic, or where conditions like "paracompact" or "complete metric space" or "second countable Hausdorff" are implicit. So either:

The wikipedia entry has a mistake (I would want to sanity-check this before editing the entry, because this is such a fundamental definition which thousands must have read).
Somewhere in that definition, the condition that $M$ be paracompact is implicit.

Question: Should the definition above indeed require that $M$ be second-countable Hausdorff or paracompact or whatever? Or is it somehow implicit somewhere, and if so, where?

Also, is this definition given carefully in any textbook?
Update: I have editted the Wikipedia article to require that $M$ be second-countable Hausdorff. But I'm still wondering if there is a textbook covering this stuff, and whether requiring the sheaf to be acyclic might have worked instead as an alternative.

REPLY [11 votes]: 1) Godement has written a book on Lie groups where he defines and uses manifolds through sheaves.
This is not surprising since he wrote a treatise on sheaf theory nearly sixty years ago, which surprisingly is still the standard reference on the subject.
The sheaf theory is very easy, since the structural sheaf is a ring of functions and thus automatically separated (= satisfies first axiom for a presheaf to be a sheaf). 
The book has no English translation (to my knowledge), but if you can overcome that hurdle you will be able to savour Godelment's inimitably idiosyncratic style, as well as the  expertise of this great mathematician .  
2) Since you mention acyclicity, let me remark that it does not follow from  either definition but is a theorem.
It is a consequence of the existence of partitions of unity, which implies that the structural sheaf $\mathcal C^k_M$ is fine, hence acyclic.
However partitions of unity require $M$ to be paracompact, which might be an argument for including paracompactness (or equivalent conditions) in the definition.