Let $X$ be a path-connected Hausdorff space and $p\in X$ a point that has an open neighborhood homeomorphic to $\mathbb{R}^d$ for some $d\geq3$. Show that for any $x_0\in X\setminus\{p\}$, we have $\pi_1(X,x_0)\cong\pi_1(X\setminus\{p\},x_0).$
I have an idea to prove this statement, but it seems a bit too straightforward, so I’d like some checking.
For any loop $\gamma$ that passes through $p$ in $X$, if I can find a loop $\alpha$ that avoids $p$ and homotopic to $\gamma$, then I’m done. Let $U$ be the open neighborhood of $p$ that is homeormorphic to $\mathbb{R}^d$. Every time a portion of $\gamma$ enters $U$, we just replace it with a homotopic path that avoids $p$, which is possible in $\mathbb{R}^d$. $\alpha$ would then consists of the portions of $\gamma$ outside $U$ and those substituting sub-paths.
If this is indeed the idea, what’s the role of the Hausdorff property of the space?