A topologist can mould a clay coffee mug into a doughnut without tearing or gluing — the handle becomes the hole. If one shape can stretch smoothly into another, they're homeomorphic: the same shape as far as topology cares.
Rubber-sheet geometry
Topology studies properties preserved under continuous deformation. Two spaces are homeomorphic if there is a continuous bijection with a continuous inverse between them — intuitively, if one can be stretched or bent (but not cut or glued) into the other. Homeomorphic spaces are 'the same' topologically.
A mug and a torus are homeomorphic: each has exactly one hole, and you can deform one into the other without cutting. A sphere, with no hole, is not homeomorphic to either.
A square and a circle are homeomorphic — bend the corners round. But a circle and a line segment are not: removing one interior point disconnects the segment differently than the circle.
- Compare a solid disk and a sphere's surface.
- Ask whether one can deform into the other without cutting or gluing.
- Use the number of holes to decide.
What you should see: They are not homeomorphic: the disk has a boundary and no hole, the sphere's surface has neither boundary nor hole — different invariants, so not equivalent.