I wonder how to prove rigorously that the figure eight, or a pair of circles intersecting at one point (looking like OO), is homotopy equivalent to a disjoint pair of circles joined by a straight line segment (looking like O-O)?
On an intuitive level, I can define a map from O-O to OO by contracting the line segment, and I can map OO to O-O by mapping the left-hand circle into itself and the right-hand circle into -O. But I have no idea how to prove that the compositions of those maps are homotopic to constant maps (and it's even not intuitively clear to me). I can trace what the compositions are, but in order to prove it rigorously I guess I have to introduce some kind of coordinates and write down the maps explicitly, which I have a hard time doing.
So what's the easiest way to show that the two spaces are homotopy equivalent? If my maps are not optimal, feel free to suggest other maps for which it's easier to prove that their compositions are homotopic to the identity. Or does the existence of a homotopy equivalence follow from some more general fact?