homeomorphism

A homeomorphism is topology's notion of “the same shape”. If you can deform one space into another by stretching and bending without cutting or gluing — and undo the deformation just as smoothly — the two spaces are homeomorphic and, as far as topology can tell, identical. The proverbial coffee mug and doughnut are homeomorphic because each has exactly one hole.

Precisely, a homeomorphism is a bijection f from X onto Y that is continuous and whose inverse f-inverse is also continuous. The continuity of the inverse is essential and is not automatic: it guarantees that open sets correspond perfectly in both directions, so X and Y have literally the same open sets up to renaming points. Any property defined purely in terms of open sets — compactness, connectedness, being Hausdorff — is then shared by homeomorphic spaces.

Be careful: a continuous bijection need not be a homeomorphism. The map that wraps the half-open interval [0, 1) continuously and bijectively onto the circle is a continuous bijection whose inverse is discontinuous, because the inverse must “cut” the circle to lay it back flat. Demanding the inverse be continuous is exactly what distinguishes a genuine topological equivalence from a mere relabeling that happens to be continuous one way.