# Homotopy group of product is product of homotopy groups

From Topospaces

## Contents

## Statement

### For two based topological spaces

Suppose and are based topological spaces. Then, the following is true for the homotopy groups of the topological spaces , and the product space :

More explicitly, if and denote the projections from to and respectively, then the maps:

and:

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: