In a

topological space X, define x~y for

points x,y∈X

iff x and y are

path connected. This is

easily seen to be an

equivalence relation; the set of

equivalence classes X/~ (a

partition of X into the sets of points with

paths between them) is the set of the

*path connected components* of X.

In other words, a path connected component of X is a *maximal* (with respect to inclusion) set of points with paths to some point.

Path connected components are *not* the same as connected components, although for "simple" topological spaces they coincide. They can be considered alternative generalizations of the notion of connected components in undirected graphs.