I have just begun to learn about the fundamental group. An exercise asks me to prove that $X=\{(x,y,z): z \ge 0\}-\{(x,y,z): y=0,0\leq z \leq 1\}$ has trivial fundamental group.
What I know is:
1) the definition of the fundamental group.
2) X has trivial fundamental group iff any loop in X can be shrunk into a constant loop at the base point.
3) Homeomorphic (path-connected) spaces have isomorphic fundamental groups.
4) Any convex subset of $\mathbb{E}^n$ and $S^m,m\ge 2$ has trivial fundamental group.
I tried to construct a homeomorphism from X to a convex subset of $\mathbb{E}^3$ such as an area like this: $\{(x,y,z): -1\leq y \leq 1,z>0\}$ But I failed.
Can you please help me? Thank you!