Consider two-dimensional surfaces. You can squeeze a sphere into a dumbbell and the topology is preserved (just concentrate on the exterior "skin" of the object, not on its volume). A sphere and a dumbbell have no holes; thus both are called simply connected spaces. If we allow for holes, we change the topology.

Both a doughnut and a coffee mug with a handle have just one hole. Thus they share the same topology: You could take a doughnut made of clay and smoothly deform it into a mug. But tear the surface of the clay to make a hole in the bottom of the mug, and you change the topology from that of a doughnut to that of an eyeglass frame.