homeomorfi

Homeomorfi, or homeomorphisms, are the central notion of topological equivalence. Two topological spaces X and Y are called homeomorphic if there exists a bijective function f: X -> Y that is continuous and whose inverse f^{-1}: Y -> X is also continuous. Such a map preserves the topological structure, so X and Y have the same topological properties.

Key properties are that the composition of homeomorphisms is a homeomorphism, and the inverse of a homeomorphism is a homeomorphism. Being homeomorphic is an equivalence relation on the class of topological spaces, partitioning spaces into homeomorphism classes: spaces in the same class are, from a topological viewpoint, the same.

Examples help illustrate the concept. The identity map on any space is a homeomorphism. In Euclidean space, translations, rotations, and reflections are homeomorphisms. The exponential map exp: R -> (0, ∞) is a homeomorphism with inverse the natural logarithm log. The open interval (0,1) is homeomorphic to the real line R, for instance via the map x -> tan(pi x - pi/2). Non-examples include maps that are bijective but not continuous (or whose inverse is not continuous) and spaces that differ in a topological invariant, such as compactness: [0,1] is compact while (0,1) is not, so they are not homeomorphic.

Applications and significance: identifying homeomorphisms establishes when spaces are topologically the same, guiding classification and understanding of which properties are preserved under continuous deformations (without tearing or gluing). They provide a rigorous language for comparing geometric objects up to intrinsic shape rather than exact form.