A proper understanding of black holes requires that we look at general relativity. General relativity tells us not only that everything within a certain radius is drawn irretrievably (we think) into it, but that the * very space and time* in the vicinity are seriously distorted. In fact, that can be taken as the very definition of gravity. In particular, time runs slower (as seen by an outside observer) near the black hole, and stops entirely at the event horizon. Space is also distorted -- I believe that a ruler in the direction of the gravity is shortened (as seen by an outside observer?) (Maybe someone can clarify this). The implications of this are that the laws of geometry as we ordinarly know them break down near a black hole. In particular, it would seem that the distance from one side of a black hole to the other (that is, from one side to the other of a mathematical sphere located ouside of the event horizon), as measured by laying a tape measure from the one side to the other through the black hole, is not equal to the circumfrence divided by pi, but is something quite different, possibly infinity, much in the same way as when you measure from one side of the mouth of a mine shaft to the other, by laying the ruler along the walls of the shaft, and find that the distance is far greater than the conventional distance acress the mouth. If that is so, then we would be justified in declaring that a black hole is indeed a hole, geometrically speaking, in every sense of the word. Perhaps it is a tunnel leading off into some other dimension.

Related to this question is what happens to the geometry of a sphere surrounding the event horizon. Normally, the laws of spherical geometry state that the integral over the surface of the sphere, of the product of the eigencurvatures of each area element, times the size of the element, equals 2 pi steradians. But when a black hole is inside the sphere, that may not be true any more.