Homotopy group of product is product of homotopy groups
For two based topological spaces
More explicitly, if and denote the projections from to and respectively, then the maps:
then under the isomorphism we get the direct factor projections for the group product.
For two topological spaces without basepoint specification
Suppose and are both path-connected spaces, or more generally, each of them is a space such that all the path components of the space are homeomorphic to each other. Then, the fundamental groups , , and are all well-defined without specification of basepoint. We then have: