Homeomorphisms

In topology, a homeomorphism between topological spaces X and Y is a bijective function f: X → Y that is continuous and whose inverse f^{-1}: Y → X is also continuous. Such a map is called bicontinuous.

Homeomorphisms are the isomorphisms of the category of topological spaces; X and Y are called homeomorphic when there exists a homeomorphism between them. The relation is an equivalence relation: reflexive, symmetric, and transitive, since the identity map is a homeomorphism and the composition of homeomorphisms is a homeomorphism. Homeomorphisms preserve topological structure, so properties such as connectedness, compactness, and being Hausdorff are invariant, as are many other invariants like the number of connected components and convergence behavior.

Examples include: the open interval (0,1) is homeomorphic to the real line R; a circle S^1 is homeomorphic to any simple closed curve in the plane (e.g., an ellipse); the unit interval [0,1] is not homeomorphic to (0,1) because [0,1] is compact while (0,1) is not. A square and a disk are homeomorphic, since a continuous deformation can map a square onto a disk without tearing.

In practice, determining whether two spaces are homeomorphic is the central task of topology's classification problems, and many invariants are used to distinguish non-homeomorphic spaces. A homeomorphism induces isomorphisms on fundamental groups and other algebraic invariants; it also preserves dimension in appropriate senses.