This is, quite possibly, my favorite pieces of math of all time. As I say in the video, I think it was the first time I realized why mathematicians care about the properties that certain shapes have which don't change as you squish and morph around that shape. Sure, the mobius strip, torus and Klein bottle are mildly interesting to look at, but until you see one used to solve a problem, contemplating these shapes feels more like an intellectual indulgence than math.

I never actually define topology in the video, doing so only really makes sense if you have some history with more abstract math, but I do think the viewer who understands the cleverness here has an appreciation for the spirit of topology itself.