Here what i want to show is $B^3 \setminus T^2$, $R^3 \setminus S^1$, $S^2 \bigvee S^1$, $i.e$, three spaces are deformation retract to each other.
Can you give me some hints or concept(?) geometric way to show this?
I found some relevant materials on online. see problem-6.
Ah, my question might be wrong. I should say $S^1$ is embedded in $R^3$ Taking A to be the embedding of $S^1$ in $R^3$ and $X = R^3-A$, X is deformation retractable. As stated in problem 6.