A homology class in singular homology is represented by a homology cycle, a finite
formal linear combination of continuous
maps from standard simplices to the space,
with zero boundary.
Two homology cycles represent the same homology
class when their difference is a homology boundary.
For instance, a path from a point 
to a point 
has boundary 
, so any two points
in a pathwise-connected component are homologous
and represent the same homology class.
