A hole in a mathematical object is a topological structure which prevents the object from being continuously shrunk to a point.
Thursday, 27 December 2012
Philosophers have sometimes debated whether there are, or aren't, holes. I say there are holes, and I once spent a while thinking about how one would try to define "hole" adequately. The conclusion I came to was based on the notion of a hole in a manifold---roughly, a region has a hole in it if that region cannot be smoothly contracted to a point. I hadn't thought about it for ages, but pleasingly, this is the definition given by Eric Weisstein at Wolfram Mathworld: