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4 added 766 characters in body

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.''

• 3 edited body

To ensure the Hausdorff condition, you can say that global sections of your sheaf separate points.

Paracompactness can be stated by saying that the structure sheaf is fine.

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 + countably many connected components implies second countability.

And the condition that $M$ has only countably many connected components can be reformulated by saying that $C^{\infty}(M)$ has only countably many indecoposable idempotents (an idempotent $p\neq 0$ in a commutative ring $A$ is called indecomposable if

$$p=q +r; r^2 =r; q^2 =q , q \neq 0 \Rightarrow q p = q$$

holds. Indecomposable idempotents 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:

1. 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.
2. The global sections of $C^{\infty}(M)$ separate points.
3. The structure sheaf $C^{\infty}$ is fine.
4. $C^{\infty}(M)$ has at most countably many indecoposable idempotents.''

However, I think that sheaf theory and locally ringed spaces are the wrong software for differential geometry and differential topology.

2 added 1389 characters in body

Paracompactness can be stated by saying that the structure sheaf is fine.

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 + countably many connected components implies second countability.

And the condition that $M$ has only countably many connected components can be reformulated by saying that $C^{\infty}(M)$ has only countably many indecoposable idempotents (an idempotent $p\neq 0$ in a commutative ring $A$ is called indecomposable if

$$p=q +r; r^2 =r; q^2 =q , q \neq 0 \Rightarrow q = q$$

holds. Indecomposable idempotents 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 $x \in M$ 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 $C^{\infty}(M)$ separate points.
• The structure sheaf $C^{\infty}$ is fine.
• $C^{\infty}(M)$ has at most countably many indecoposable idempotents.''
• However, I think that sheaf theory and locally ringed spaces are the wrong software for differential geometry and differential topology.

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