Let $D^2=\{x\in\mathbb{R}^2:\|x\|\le1\}$, $x\neq a\in D^2$. Find $\pi_1(X\setminus\{x\},a)$ if:
a. $x\in\partial D^2$
b. $x\in \text{int} D^2$
about the first one I think the fundamental group is trivial (means one singleton $\{a\}$) but I can't figure out how can I prove it.
about the second, it seems clear that after removing point from the interior, it's homotopic to $S^1$ (which its fundamental group is known) but I can't find (for example) a retraction which sends the disc with hole in the interior to $S^1$.
So which homomorphism between $\pi_1(D^2\setminus\{x\})\to\mathbb{Z}\setminus\{0\}$ can I pick ? how can I define a deformation retraction between $D^2\setminus\{x\}$ where $x\in\text{int}D^2$ and $S^1$?
(a) Notice that your region is contractible (in fact, starlike).
(b) Note that $S^1$ can be identified with the boundary of $D^2$. So, we can can imagine a homotopy from $D^2 - \{x\}$ to $S^1$ by pulling wider the "hole" at $x$; one way to do precisely is by projecting radially from $x$ onto $S^1$.