I'm going through the crisis of being unhappy with the textbook definition of a differentiable manifold. I'm wondering whether there is a sheaf-theoretic approach which will make me happier. In a nutshell, is there a natural condition to impose on a structure sheaf $\mathcal{C}^k_M$ of a topological space $M$ that can stand-in for the requirement that $M$ be second-countable Hausdorff?

<h3>Background</h3>

Most textbooks introduce differentiable manifolds via <a href="http://en.wikipedia.org/wiki/Differentiable_manifold#Definition">atlases and charts</a>. This has the advantage of being concrete, and the disadvantage of involving an arbitrary choice of atlas, which obscures the basic property that a differential manifold "looks the same at all points" (diffeomorphism group acts transitively). And isn't introducing local coordinates "an act of violence"?

I saw a much nicer definition of differentiable manifolds on Wikipedia, which I don't know a good textbook reference for. This definition proceeds via <a href="http://en.wikipedia.org/wiki/Differentiable_manifold#Sheaves_of_local_rings">sheaves of local rings</a>. The Wikipedia definition stated:

<blockquote>
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 $\mathbb{R}$-algebras defined on $M$, such that the locally ringed space $(M,\mathcal{O}_M)$ is locally isomorphic to $(\mathbb{R}^n, \mathcal{O})$. [$\mathcal{O}(U)=C^k(U,\mathbb{R})$ is the structure sheaf on $\mathbb{R}^n$.]
</blockquote>

Beautiful, really! Entirely coordinate free. But isn't there a General Topology condition missing?

I <a href="http://math.stackexchange.com/questions/107319/sanity-check-about-wikipedia-definition-of-differentiable-manifold-as-a-locally">confirmed on math.SE</a> (to make sure that I wasn't hallucinating) that this definition is indeed missing the condition that $M$ be second-countable Hausdorff. That indeed turned out to be the case, so I edited the Wikipedia definition to require $M$ to be second-countable Hausdorff.<br>

<h3>Why am I still not happy?</h3>

The deep reason that we require a differentiable manifold to be paracompact, as per <a href="http://math.stackexchange.com/questions/98105/why-are-smooth-manifolds-defined-to-be-paracompact/98124#98124">Georges Elencwajg's extremely informative answer</a>, is that paracompactness makes sheaves of $C_M^k$-modules (maybe $k=\infty$) <a href="http://en.wikipedia.org/wiki/Injective_sheaf#Acyclic_sheaves">acyclic</a>. This is a purely sheaf-theoretic property (a condition on the structure sheaf of $M$ rather than on $M$ itself), which quickly implies good things like that every subbundle of a vector bundle on $M$ be a direct summand. Is this in fact enough?

If it were enough to require that $\mathcal{O}_M$ be <a href="http://en.wikipedia.org/wiki/Injective_sheaf#Acyclic_sheaves">acyclic</a>, or maybe <a href="http://en.wikipedia.org/wiki/Injective_sheaf#Fine_sheaves">fine</a>, then the nicest, most flexible (and, in a strange sense, most enlightening) definition of differentiable manifold, would be:

<blockquote>
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 an <b>acyclic</b> sheaf of local $\mathbb{R}$-algebras defined on $M$, such that the locally ringed space $(M,\mathcal{O}_M)$ is locally isomorphic to $(\mathbb{R}^n, \mathcal{O})$. 
</blockquote>

Maybe the word <b>acyclic</b> should be <b>fine</b>.

<blockquote>
<b>Question</b>: Can I put a natural sheaf-theoretic condition on $\mathcal{O}_M$ (acyclic? fine?) which ensures that $M$ (a topological space) must be a second-countable Hausdorff space? If not, would such a condition at least ensure that $M$ be a generalized differentiable manifold in some kind of useful sense? 
</blockquote>