# Defining a Manifold by Sheaves

1. Jun 8, 2014

### 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?

2. Jun 8, 2014

### micromass

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

3. Jun 8, 2014

### Mandelbroth

Alright. That makes more sense. Thank you!

4. Jun 8, 2014

### micromass

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