fermës

Fermés is the plural form of the French adjective fermé, meaning closed or shut. In mathematical contexts, fermés denotes closed sets in a topological space. A subset F of a topological space X is called fermé if its complement X \ F is open. Equivalently, a subset is fermé if it contains all of its limit points, or if it equals its closure.

Key properties include that the whole space X and the empty set ∅ are fermés; the intersection of any collection of fermés is fermé; and the union of finitely many fermés is fermé. The concept is dual to that of open sets: a set is fermé if and only if its complement is open.

In metric spaces, a set is fermé precisely when it contains all its limit points; this is often checked by verifying that every convergent sequence in the set has its limit also in the set. Another common characterization is that a set is fermé if it equals its closure, and the closure of any set S is the smallest fermé set containing S.

Examples in the real line with the standard topology include closed intervals [a,b], finite sets like {x}, and more generally any set that contains all its boundary points. The open intervals (a,b) are not fermés. If f is a continuous function between topological spaces and C is fermé in the codomain, then the preimage f^{-1}(C) is fermé in the domain.

Fermés play a central role in analysis and topology, underpinning concepts such as convergence, continuity, and compactness, and they interact closely with open sets through complementation and closure operations.