Fundamental group of some spaces

Question

Which is the fundamental group of:

a) $X=\{(x,y,z) \in \mathbb{R}^3: \, x^2+y^2-z^2=0, z>0\}$; b) A cylinder minus a point; c) A triangle with the three vertices identified.

My answers are:

$a) \pi_1(X)=(0)$,

$b) \mathbb{Z}$,

$c) \mathbb{Z}*\mathbb{Z}*\mathbb{Z}$;

but I need a formal proof. Can you help me?

Answer 1

I am sorry that you are right,$X*I$ is a cylinder,it is $CX=X*I/X*{1}$,a quotient topology space,The top point is its retract,so fundamental group is trivial.(Also its first homology group trivial).

I am sorry for my misguiding.