Topology extends the notion of a
connected component in an
undirected graph (described above). The connected components of a
topological space X are the
maximal (by inclusion) connected
subspaces of X.
Equivalently: Define an equivalence relation x~y ("x and y are connected") on points in X: x~y iff there is no partition X=A∪B into disjoint open subsets A,B for which x∈A and y∈B. The connected components of X are the equivalence classes of ~ (aka "X/~").
A related notion is path connected component; however, due to the difference between connectedness and path connectedness, the two concepts are not equivalent for general topological spaces.