question regrading double cone

Question

How to prove that double cone is not a topological manifold.I think I have to do something with the meeting point of the two cones.

Answer 1

Hint: By observation, the space around $(0,0)$ is giving us problems. In fact, it looks like 2 copies of $\mathbb{R}^2$ glued at a point.

Using this fact, show that any neighborhood around $(0,0)$ is not homeomorphic to $\mathbb{R}^2$ (and clearly not $\mathbb{R}^n$ for any other $n$), by considering the image of the lower cone (without the apex) and the upper cone (without the apex), which are 2 disjoint open sets.

Hence, it is not a topological manifold.