Why aren't two cones attached in their vertex a manifold?

Question

In this general relativity lecture notes, on page two, it is claimed that two cones attached on their vertex do not constitute a manifold. I don't see why. So, why?

Answer 1

Recall that a manifold is locally homeomorphic to Euclidean space. Think about an X-shape, with $p$ the cross point; locally around $p$, the X-shape does not look like Euclidean space, even though around every other point it is locally a line.

Answer 2

Any connected neighborhood of the vertex becomes disconnected when the vertex is removed. The only Euclidean space in which such behavior is possible is the real line. And, clearly, the space you are trying to describe is not locally homeomorphic to the real line in general, since the veetex is the only point where this behavior happens.