> I ask because a circle around a cusp in a sheet will get “caught” on the infinite spike; and a circle not around the cusp will end up with weird deformations

> I ask because a circle around a cusp in a sheet will get “caught” on the infinite spike; and a circle not around the cusp will end up with weird deformations if it comes close. That suggests there’s something “topological” about such a cusp, akin to there being a hole. Your intuition is right, this is an example of a topological defect, but it's not directly related to circles having the "wrong" sizes. It's topological in that it's just a property of how your sheet is connected, and can't be changed by continuous deformations. Black holes are not topological defects in the same way, because they're not really "infinite spikes".