Fundamental group of $X=\{(x,y,z) \in \mathbb{R}^3: \, z^2=x^2+y^2, \, z >0\}$

Question

Which is the fundamental group of $X=\{(x,y,z) \in \mathbb{R}^3: \, z^2=x^2+y^2, \, z >0\}$? $X$ is the cone without the vertex. Is it homotopic to $S^1$?

Answer 1

The projection along $z$ is a homeomorphism $X\cong \mathbb{R}^2-\{pt\}$. $\mathbb{R}^2-\{pt\}$ is homotopic to $S^1$. Therefore $\pi_1(X) = \mathbb{Z}$.