Define the fundamental group of a topological space and explain briefly why a continuous map gives rise to a homomorphism between fundamental groups.

Let $X$ be a subspace of the Euclidean space $\mathbb{R}^{3}$ which contains all of the points $(x, y, 0)$ with $(x, y) \neq(0,0)$, and which does not contain any of the points $(0,0, z)$. Show that $X$ has an infinite fundamental group.