[brianpan]

Have you heard the quip that in physics a cow and a point are equivalent? This is because the physicist cares only about the motion of the thing.

In topology, a doughnut and a coffee mug are equivalent (a mug has exactly one hole, in the handle where your fingers grab). Because the mathematician doesn't care about how hard, thick, or breakable it is; they only care about how complex the shape is. So throw out thickness, size, elasticity, etc.