Fascinating! This idea of an inner hole that is hidden and connected is quite unintuitive to me. I majored in mathematics in college but I wouldn't have reached this conclusion through intuition.
What do you mean by the hole in the torus being "hidden and connected"?
As an ant you could easily see that you were not on the surface of an infinite plane simply because you keep coming back to the same spot over and over again. In order to figure out that there was a hole (or more appropriately named as 'handle'), you'd need to leave some thread behind and notice that there was no way to contract various loops to a point.
In a torus, there are effectively two holes -
1. the center hole around which there is a cylindrical ring
2. the whole inside the cylindrical ring, which is hidden and connected.
When we cut along the length of the cylindrical ring, we are effectively creating two edges, just like if we were to cut a circular ring of wire, we would end up with two points at the end.
The two edges then open up, effectively forming another cylinder, which has to be cut again.
Any other object which has only a single hole, would require only a single cut to flatten out.
(a) a SURFACE generated by a circle rotated about a coplanar axis outside the circle. Examples: an inner tube or a swimming tube. (This is what's meant by "a hollow torus" in the description).
(b) a SOLID generated by a circular lamina rotated about a coplanar axis outside the lamina. Example: a bagel or an annular gasket. I was thinking of this one, and the descriptions made no sense at all.
Now it all makes sense.
Hmm. I thought before this that I knew what a torus was...