OK, yes. I can tell you get it because of your description of a fork. And the idea of an intrinsic characteristic is a very mathematical way of thinking about it.
The hole, in fact, is intrinsic. Here’s another way of saying something doesn’t have a hole: pretend your thing is made of a stretchy flexible material. If you can deform it into a sphere without breaking the surface at all, then it does not have a hole. You can’t do this with a donut. If you cut it a bit you could straighten it out into a cylinder and then squish it into a ball. But without cutting it, you couldn’t get rid of the hole.
So a donut has a hole, and a fork, which can be deformed into a ball, does not. This is whence comes the joke that a topologist is someone who cannot distinguish her coffee cup from a donut. (because of the hole through the handle).
