homöomorph

A homöomorph, or homeomorphism, is a central concept in topology that describes a structure-preserving equivalence between topological spaces. Formally, if X and Y are topological spaces and f: X → Y is a function, f is a homöomorph if it is bijective, continuous, and its inverse f⁻¹: Y → X is also continuous. Equivalently, f is bicontinuous, meaning open sets map to open sets and closed sets map to closed sets in both directions. If such an f exists, X and Y are called homeomorphic, and they are considered the same from a topological viewpoint.

Homeomorphisms preserve fundamental topological properties, such as connectedness, compactness, and the Hausdorff condition, as well as more refined invariants like dimension and genus in appropriate contexts. Examples: the identity map on any space is a homeomorphism; rotations of the unit circle S¹ are homeomorphisms; a map from the open interval (0,1) to the real line R given by x → tan(π(x − 1/2)) is a homeomorphism, showing (0,1) is topologically the same as R. In contrast, a constant map or any continuous bijection whose inverse is not continuous cannot be a homeomorphism; spaces with different basic properties (for instance, the non-compact line R and the compact circle S¹) are not homeomorphic.

The collection of all homeomorphisms from a space X to itself forms the homeomorphism group Homeo(X), under composition. In topology, classification often proceeds by determining whether spaces are homeomorphic, treating homeomorphism as the notion of equivalence.