Hurewicz map is well-defined
The strict map
The map is defined as follows. First define a map:
which essentially uses the identification of with the quotient of by the collapse of its boundary to a single point, i.e., a homeomorphism .
Now given any based continuous map , consider . This gives a -singular chain in , and its homology class is precisely the element we are looking for.
What we need to show
To note that this is indeed well-defined, we need to show that if and are homotopic maps as based continuous maps from to , then and are both in the same homology class.
Fill this in later