In a topological space
X, define x~y for point
s x,y∈X iff
x and y are path connected
. This is easily
seen to be an equivalence relation
; the set of equivalence class
es X/~ (a partition
of X into the sets of points with path
s between them) is the set of the path connected components
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.