A homeomorphism is a bijective continuous map f: X → Y whose inverse f⁻¹: Y → X is also continuous. Two topological spaces are homeomorphic if there exists a homeomorphism between them — they have exactly the same topological structure, differing only in the labels of their points.
A homeomorphism establishes a one-to-one correspondence between the open sets of X and Y: V is open in Y if and only if f⁻¹(V) is open in X. Any topological property — connectedness, compactness, number of connected components, fundamental group — is preserved by homeomorphism. Properties preserved by homeomorphism are called topological invariants, and finding complete invariants is a central problem in topology.
Continuity of f alone does not guarantee a homeomorphism: the map wrapping an open interval onto a circle is continuous and bijective, but its inverse is not continuous. Requiring both f and f⁻¹ to be continuous ensures that the topological structures match in both directions.
Homeomorphism is the correct notion of isomorphism in the category Top of topological spaces. In homotopy theory, a weaker notion — homotopy equivalence — allows spaces to be identified even when they are not homeomorphic, provided they have the same “shape” up to continuous deformation. The passage from homeomorphism to homotopy equivalence is the passage from topology to homotopy theory, and ultimately to ∞-categories.