Suppose we have a circle and a line in $\mathbb{R}^3$. The line lies inside the circle without intersecting it. Consider the space $X$ obtained from $\mathbb{R}^3$ minus this line and this circle. I know this space should retract to a torus $S^1 \times S^1.$
What I think is the following.
If I take a plane where the line lies on, this plane intersects the circle in two points.
The intersection of this plane with $X$ should be $\mathbb{R}^2$ minus three points. This work for any of the infinite planes where the line lies on.
So the idea would be to find a retraction onto some intersection of the torus with a plane for every such plane and then glueing those together.
Is it possible to conclude with this line of reasoning?