Question

To ensure

Definition: ''A smooth manifold is a locally ringed space $(M;C^{\infty})$ which satisfies the Hausdorff condition, you can say conditions:

Each $x \in M$ admits a neighborhood $U$, such that $(U,C^{\infty})$ is isomorphic to $(\mathbb{R}^n,C^{\infty})$ as a locally ringed space.

The global sections of your sheaf $C^{\infty}(M)$ separate points.

Paracompactness can be stated by saying that the

The structure sheaf $C^{\infty}$ is fine as a sheaf of modules over itself.

Paracompactness and second countability are not equivalent conditions on locally euclidean Hausdorff spaces (second countability implies paracompactness, but not vice versa, see

$\mathbb{R}$ with the discrete topology). But paracompactness + C^{\infty}(M)$ has at most countably many connected components implies second countabilityindecoposable idempotents.''

Explanations:

1) is evident.

And

2) means that for $x \neq y \in M$, there exists $f \in C^{\infty}(M)$ with $f(x)=0$, $f(y) \neq 0$. This ensures the Hausdorff condition once we know that elements of $C^{\infty}(M)$ give rise to continuous maps $M \to \mathbb{R}$. This is as follows: $f \in C^{\infty}(M)$ given, $x \in M$has only countably many connected components . Pick a chart $h:U \to \mathbb{R}^n$; under this chart, $f|_U$ corresponds to a smooth function on $\mathbb{R}^n$, whose value at $h(x)$ does not depend on the choice of the chart. Call this value $f(x)$. Checking the continuity of $x \mapsto f(x)$ can be reformulated by saying done in charts.

3.) By this I mean that for each open cover $C^{\infty}(M)$ has only countably many indecoposable idempotents (U_i)$, there is a partition of unity $\lambda_i$ with the usual properties and that $\lambda_i$ is a map of $C^{\infty} (an M)$-modules. A standard argument shows that $\lambda_i$ is given by multiplication with a smooth function. Therefore, the underlying space $M$ is paracompact.

4.) An idempotent $p\neq 0 $ in a commutative ring $A$ is called indecomposable if

holds. Indecomposable idempotents in $C^{\infty}(M)$ correspond to connected components.

So here is the definition:

Definition: ''A smooth manifold is a locally ringed space $(M;C^{\infty})$ which satisfies the conditions:

Each Therefore condition 4 means that $x \in M$ admits a neighborhood $U$, such has only countably many connected components.

These conditions together imply that $(U,C^{\infty})$ M$ is isomorphic to $(\mathbb{R}^n,C^{\infty})$ as Hausdorff and second countable, because a locally ringed euclidean, connected and paracompact Hausdorff space .

The global sections is second countable, see Gauld, "Topological properties of $C^{\infty}(M)$ separate pointsmanifolds", Theorem 7 (see http://www.jstor.org/stable/2319220 ).

The structure sheaf Paracompactness alone does not guarantee second countability, see $C^{\infty}$ is fine\mathbb{R}$ with the discrete topology.

$C^{\infty}(M)$ has at most countably many indecoposable idempotents.''