This blog post is on the behavior of homotopy groups with respect to products. Proposition 4.2 of Hatcher:

For a product of an arbitrary collection of path-connected spaces there are isomorphisms for all .

The proof given in Hatcher is a short one: A map is the same thing as a collection of maps . Taking to be and gives the result.

A possible alternative proof is to first prove that , which is the result for a product of two spaces. The general result then follows by induction.

We construct a map , .

Notation: , , where are the projection maps.

We can show that , thus is a homomorphism.

We can also show that is bijective by constructing an explicit inverse, namely , where , .

Thus is an isomorphism.