Let $X$ be a path-connected topological space, $x_0\in X$, s.t. $x_0$ has an open neighbourhood homeomorphic to $\mathbb{R}^d$ for some $d\ge 3$.
The task is to show that $\pi_1(X)\cong \pi(X\setminus\{x_0\})$ (the base point does not matter as all spaces in question are path-connected).
We may use that $X\setminus\{x_0\}$ is path-connected. I have already shown the claim in the case that $\{x_0\}$ is closed, however I haven't been able to proof it without assuming this.