Homology of $\mathbb{R}\setminus A_+$.

Question

Let $A$ be the unit circle in the $xy$ plane in $3$-dimensional real space and let $A_+$ be a semicircle. I have to compute the homology of $\mathbb{R}^3\setminus A_+$. I was thinking that $\mathbb{R}^3\setminus A_+$ is homotopically equivalent to $\mathbb{R}^3\setminus \{(1,0,0)\}$ which can than be proved to be homeomorphic to $S^2$. However, I have trouble finding a retraction $g:\mathbb{R}^3\setminus \{(1,0,0)\}\rightarrow \mathbb{R}^3\setminus A_+$ to construct the equivalence, since homotopy equivalence is not stable under subtractions, i.e. if $A,B\subset X$, $A$ homotopically equivalent to $B$ it does not follow that $X\setminus A$ homotopically equivalent to $X\setminus B$.