I have a mountain floating in space. Topologically, it's a sphere.
I put a cave in the mountain. Topologically, it's still a sphere.
I add a stalactite (a cone of rock hanging from the roof of the cave.) Still a sphere.
The stalactite grows to join the floor of the cave; the mountain no longer deforms to a sphere.
(An alternative visualization: touch your fingers to your thumb, as if holding an imaginary pole. Dip your hand in liquid rubber up to the wrist. Let the rubber dry and remove your hand. What's the shape of the rubber?)
It's not a torus; that has a hole that goes from an entrance to an exit; but this has one entrance only.
It's not a double torus; that has two distinct holes, like a 3-dimensional "B".
I've found a lot of fascinating equations for n-holed toruses, but nothing that seems equivalent to my object. It doesn't seem to be a punctured torus, though maybe I'm looking for the solid of which the punctured torus is the (partial) surface.
Does my object have a name?
