Voisinage

Voisinage is a term used in mathematics, especially in topology, to denote the neighborhood of a point. A set V is called a neighborhood (un voisinage) of a point x in a topological space X if there exists an open set U with x in U and U contained in V. Equivalently, V is a neighborhood of x if it contains an open set that contains x. If V itself is open and contains x, V is an open neighborhood of x. For a fixed x, the collection N(x) of all neighborhoods of x forms a filter that encodes the local structure around x.

In metric spaces, neighborhoods are often described via open balls: B(x, r) = {y : d(x, y) < r}. Any set that contains such a ball is a neighborhood of x; conversely, every neighborhood contains such a ball for some r > 0. The concept generalizes to nets and sequences: a sequence x_n converges to x if, for every neighborhood V of x, x_n eventually lies in V.

The notion of voisinage plays a central role in definitions of continuity and limits. A function f is continuous at x if for every neighborhood V of f(x), there exists a neighborhood U of x such that f(U) ⊆ V. Equivalently, behavior near x is described in terms of neighborhoods around x and f(x).

Beyond mathematics, the word voisinage also appears in everyday French to denote the surrounding area or vicinity of a place—the environment or locality around it.