Ladhérence

L’adhérence, written as l’adhérence in French and often translated as closure in English, is a fundamental concept in topology. Given a topological space X and a subset A, l’adhérence of A is the smallest closed set that contains A; equivalently, it consists of all points x in X such that every open neighborhood of x intersects A. This includes A itself and all its limit points.

The closure of A is usually denoted cl_X(A) or simply cl(A). It is closed by definition, contains A, and is idempotent: cl(cl(A)) = cl(A). Basic laws include cl(A ∪ B) = cl(A) ∪ cl(B) and, if A is already closed, cl(A) = A. If A is dense in X, then cl(A) = X.

A standard way to understand closure in metric spaces is through limit points: a point x belongs to cl(A) if and only if every neighborhood of x contains a point of A. In metric terms, this is equivalent to the distance from x to A being zero. The closure operation also relates to the interior and boundary of a set: the boundary ∂A equals cl(A) \ int(A), where int(A) is the largest open set contained in A.

Examples help illustrate the concept: in the real line with the usual topology, the closure of the interval (0,1) is [0,1]. In higher dimensions, the closure of a path-connected region includes its boundary points, capturing limits of sequences and convergent processes within the space.

In mathematics, l’adhérence is a standard tool for formalizing notions of convergence, approximation, and the persistence of properties under limits, making it central to analysis, geometry, and related fields.