I have the following problem:
Show that the cone $x^2+y^2=z^2$ with the subspace topology isn`t a manifold.
If it would be a manifold it would be $2$-dimensional. So there should be for every point in the cone a neighborhood, so that there is a homomorphism from that neighborhood to $\mathbb{R}^2$, right? But for the point $0$ there shouldn't be such a neighborhood. My problem is how to show that. Can someone help me please? Thanks in advance.
If you delete the point $0$ from a neighborhood of the cone which contains it, you're left with a disconnected set.
If you delete a point from an open set in $\Bbb{R}^2$, you are not.