prolongements

Prolongement (extension) in mathematics refers to the process of enlarging the domain of a mathematical object while preserving its defining properties. Given an object defined on a subset A of a space X, a prolongement to X is an object on X that restricts to the original on A.

Types of prolongements include: prolongement de fonctions, which means extending a function f: A -> Y to a function F: X -> Y with F|_A = f. When X is a topological space and Y is Hausdorff, uniqueness is not guaranteed; existence results include the Tietze extension theorem for real-valued continuous functions on normal spaces, and in metric spaces, uniform continuity on A implies a unique uniformly continuous extension to the closure of A. Prolongement continu refers to extension preserving continuity; this is often subtle at boundary points and requires specific conditions on A and the ambient space. Prolongement analytique denotes analytic continuation of a holomorphic function from a domain to a larger domain along analytic paths, governed by the identity theorem and possible monodromy obstructions; it is a central concept in complex analysis. Prolongement topologique concerns extension via compactifications or universal constructions, such as extending maps to a Stone–Čech or one-point compactification. Prolongement en géométrie algébrique covers the extension of morphisms defined on a generic fiber or subvariety to a model over a base scheme; common criteria involve properness, separatedness, and valuative criteria, with examples including Néron models and extensions of morphisms over discrete valuation rings.

Limitations: extensions may not exist or may not preserve properties beyond those assumed; obstructions include boundary behavior, singularities, or topological constraints. See also extension, boundary, compactification, analytic continuation, morphism, and sheaf extension.