conexe

Conexe is a term used in Romanian mathematical literature to refer to connected components of a topological space. A conexă, or connected component, is a maximal connected subset of a given space. The collection of conexe partitions the space into disjoint, connected pieces, so every point belongs to exactly one conexă—the conexa containing that point.

Formally, for a topological space X and a point x in X, the conexă of x is the set C(x) that contains x and is connected, and is maximal with respect to inclusion among connected subsets of X. Every connected subset of X is contained in some conexă, and the union of all conexe equals X. If two points lie in the same conexă, they can be joined by a chain of connected subsets staying inside that same piece.

Key properties include that conexe are pairwise disjoint and that each conexă is itself connected. The decomposition into conexe is unique, and it is invariant under homeomorphisms of the space. In metric spaces, the concept aligns with the general topological notion of connected components, though terminology may vary by language.

Examples help illustrate the idea: in the real line with the standard topology, the entire line is connected, so there is a single conexă. In a space formed by two disjoint intervals, each interval is a conexă. In the space of rational numbers with the usual topology, every conexă is a single point, since the rationals are totally disconnected.

Relation to path-connected components: a conexă need not be path-connected in general, though in locally path-connected spaces the connected components have additional structure. The concept of conexe is central to understanding the global shape and decomposition of topological spaces.