I was trying to find some basic properties of a topological space $X$ such that $\pi_1(X,x_0)=\pi_1(X-\{pt.\},x_0)$ where $pt. \neq x_0$.
For example on the sphere $S^2$ this is obviously true because $S^2-\{pt\} \simeq \mathbb{R}^2$.
So, what can we say about such spaces in general? Which are the properties required and which ones are needed?
If $X$ is a closed, connected, $n$-dimensional manifold, then the point $\{pt\}\subset X$ will admit a geodesically convex open neighborhood $N$. Hence, $N-\{pt\}\simeq S^{n-1}$ and we can apply Van-Kampen's theorem to the cover $\{N,X-\{pt\}\}$. We get $$\pi_1(X)\cong \pi_1(X-\{pt\})\ast_{\pi_1(S^{n-1})}\pi_1(N)\;.$$ In particular, if the manifold has dimension $\geq 3$ then it will alway be true that $\pi_1(X-\{pt\})\cong \pi_1(X)$, since $N$ is contractible and $\pi_1(S^{n-1})\cong 1$. If $n=2$ and the map $$i_*:\mathbb{Z}\cong\pi_1(S^1)\to \pi_1(X-\{pt\})\;,$$ induced by the inclusion sends the generator to the trivial element, then you also have an isomorphism. In particular, an orientable surface of genus $g$ will also have the property you want.