A topological space decomposes into its connected components. The connectedness relation between two pairs of points satisfies transitivity,
i.e., if 
and 
then 
.
Hence, being in the same component is an equivalence
relation, and the equivalence classes are the connected components.
Using pathwise-connectedness, the pathwise-connected component containing 
is the set of all 
pathwise-connected to 
. That is, it is the set of 
such that there is a continuous path from 
to 
.
Technically speaking, in some topological spaces, pathwise-connected is not the same as connected. A subset 
of 
is connected if there is no way to write 
with 
and 
disjoint open sets. Every topological
space decomposes into a disjoint union 
where the 
are connected. The 
are called the connected components of 
.
The connected components of a graph 
are the set of largest subgraphs of 
that are each connected.
Connected components of a graph may be computed in the Wolfram
Language as ConnectedComponents[g]
(returned as lists of vertex indices) or ConnectedGraphComponents[g]
(returned as a list of graphs). Precomputed values for a number of graphs are available
as GraphData[g,
"ConnectedComponents"].
