Connected component

Connected components are part of topology and graph theory, two related branches of mathematics.

• For the graph-theoretic concept, see connected component (graph theory).
• In topology: Define an equivalence relation ~ on a topological space X, so that we declare x~y if there is a connected subspace A of X containing both x and y. The set of all equivalence classes are called the connected components of X. (This is indeed an equivalence relation as the reader can check.) In other words, a connected component consists of all points in any connected subspace containing some given point. For example a hyperbola as a subspace of the plane has two connected components while ellipses and parabolas have only one. The narrower concept of a path-connected component refers to the set of all points which are connected by a path to some given point (see Connected space for an explanation of the difference between "connected" and "path-connected").