§ Zeroth singular homology group: Intuition
We wish to show that for a path connected space , the zeroth singular homology group is just .
The intuition is that the zeroth homology group is given by consider ,
, and then taking .
Recall that is the abelian group generated by the direct sum of generators
, where is the -simplex, that is,
a single point. So is an abelian group generated by all points in . Now, contains all paths
between all points . Thus the boundary of will be of the form . Quotienting by
identifies all points with each other in . That is, we get ,
which is isomorphic to . Thus, the zeroth singular homology group is .