Is a Manifold Defined by Sheaves Always Hausdorff?

[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.