Homeomorphism

A homeomorphism is a fundamental concept in topology that captures when two spaces have the same topological shape. Let X and Y be topological spaces. A function f from X to Y is a homeomorphism if it is bijective, continuous, and its inverse function f^{-1} from Y to X is continuous. When such an f exists, X and Y are called homeomorphic, often written X ≅ Y.

Equivalently, a homeomorphism is a continuous bijection with a continuous inverse, or a bijection that preserves open sets in both directions (an open map). Under a homeomorphism, notions defined purely topologically—such as convergence, continuity, and openness—are preserved.

Examples illustrate the idea. The real line R and the open interval (0,1) are homeomorphic. A circle S^1 in the plane is homeomorphic to any simple closed curve. The torus is homeomorphic to the product S^1 × S^1, while a sphere S^2 is not homeomorphic to a torus due to different topological invariants.

Homeomorphisms preserve many topological properties. If X is homeomorphic to Y, then X is compact if and only if Y is compact; X is connected if and only if Y is connected; homeomorphisms preserve Hausdorffness, separability, and countability properties, as well as topological dimension. In short, they identify spaces that are indistinguishable from a purely topological perspective.

What a homeomorphism does not preserve are geometric or analytic structures such as distances, angles, smoothness, or differentiable structure. A homeomorphism can be non-differentiable and can alter notions like orientation.