Defining a Manifold by Sheaves

[Mandelbroth]

While reading about sheaves, I came across a beautiful definition of a manifold. An $n$-manifold is simply a locally ringed space which is locally isomorphic to a subset of $(\mathbb{R}^n, C^0)$. However, I don't see how this guarantees a manifold to be Hausdorff. Would someone please explain this?

 

[micromass]

You need to demand Hausdorff and second countable separately since they are global conditions.

 

[Mandelbroth]

You need to demand Hausdorff and second countable separately since they are global conditions.

Alright. That makes more sense. Thank you!

 

[micromass]

By the way, if you're interested in this, check out this book:

https://www.amazon.com/dp/0821837028/?tag=pfamazon01-20

Also, it needs to be said that differential geometry doesn't really fit well in the theory of locally ringed spaces for several reasons. One thing that is very interesting is that of diffeological spaces. A diffeological space is to a differentiable manifolds as a topological space is to a topological manifold. Diffeological spaces behave way better under categorical constructions. See http://en.wikipedia.org/wiki/Diffeology The references below the wiki article are very good.

 

[blue_raver22]

Thank you for bringing up this interesting point about defining a manifold by sheaves. I would like to offer some insights on this topic.

Firstly, let's clarify the definition of a manifold. A manifold is a topological space that locally looks like $\mathbb{R}^n$, which means that every point on the manifold has a neighborhood that is homeomorphic to an open subset of $\mathbb{R}^n$. This definition does not explicitly mention the Hausdorff property, which states that for any two distinct points on a topological space, there exist open neighborhoods that do not intersect. However, this property is implicitly assumed in the definition of a locally ringed space, which is used to define a manifold by sheaves.

A locally ringed space is a topological space equipped with a sheaf of rings, where the stalk at each point is a local ring. A local ring is a commutative ring with a unique maximal ideal. This means that for any point on the manifold, there exists a neighborhood where the sheaf of rings is isomorphic to the ring of continuous functions $C^0$ on $\mathbb{R}^n$. This is where the Hausdorff property comes into play. Since the ring of continuous functions on $\mathbb{R}^n$ is a Hausdorff space, the isomorphism between the sheaf of rings and $C^0$ guarantees that the neighborhood of each point on the manifold is also Hausdorff. Therefore, the manifold itself must be Hausdorff.

In summary, while the definition of a manifold by sheaves may not explicitly mention the Hausdorff property, it is implied by the isomorphism between the sheaf of rings and $C^0$, which is a Hausdorff space. I hope this explanation helps to clarify any confusion regarding the Hausdorff property in the definition of a manifold by sheaves.