homeomorphic

Homeomorphic is a term from topology describing a strong form of geometric sameness between spaces. Two topological spaces X and Y are homeomorphic if there exists a bijective function f: X -> Y that is continuous and whose inverse f^{-1}: Y -> X is also continuous. When such a map exists, X and Y are said to be homeomorphic, often written X ≅ Y. A homeomorphism is thus an isomorphism of topological structures.

Because a homeomorphism preserves the open set structure, it maps open sets to open sets and preserves convergence, continuity properties, and many qualitative features. Consequently, homeomorphic spaces share topological properties such as connectedness, compactness, and the same dimension, while not necessarily sharing metrical properties like length or distance.

Examples illustrate the idea of topological sameness. The circle S^1 is homeomorphic to any simple closed curve in the plane, so a square, triangle, or ellipse (with boundary) is homeomorphic to S^1. The open interval (0,1) is homeomorphic to the entire real line R. A closed disk is homeomorphic to a square, since both are compact, connected 2-manifolds with boundary. A classic intuition is that a coffee mug and a torus are homeomorphic because one can deform the mug into a donut shape without tearing.

In practice, determining whether spaces are homeomorphic relies on invariants such as fundamental groups, homology, or Euler characteristic, which can distinguish spaces that are not homeomorphic. The term contrasts with isometries or diffeomorphisms, which preserve additional structure like distance or differentiable structure.