I want to compute the fundamental group of $\{x : \|x\| < 1\}$. It should be trivial as I can contract to one point. What is wrong with the following method?
Consider the map $H(x, t) = (1 - t)x + tx/(2\|x\|)$. We claim that $H$ is a deformation retraction of $X = \{x : \|x\| < 1\}$ onto $A = \{x : \|x\| = 1/2\}$. We have $H(x, 0) = x$ and $H(x, 1) \in A$ for all $x \in X$ and $H(a, t) = a$ for all $a \in A$. Since $A$ is homeomorphic to $S^{1}$, it follows that $\pi_{1}(X) = \mathbb{Z}$. What is wrong with this proof?