Topology/Singular Homology

First we define the standard simplices as the convex span of the standard basis vectors. We then take as a boundary map

Next we transport this structure to a topological space X: A simplex s in X is the image of a continous map from some standard simplex.

Now let be the free groups on the simplices in X. The maps now induce a new chain map on the complex

Now using the definition of homology as in the previous section we define (Exercise: prove that this is well-defined.)