** if **

We need the following lemma:

If a space is the union of a collection of path-connected open sets each containing the basepoint and if each intersection is path-connected, then every loop in at is homotopic to a product of loops each of which is contained in a single .

**Proof:**

Take and to be the complements of two antipodal points in . Then is the union of two open sets and , each homeomorphic to such that is homeomorphic to .

Choose a basepoint in . If then is path-connected. By the lemma, every loop in based at is homotopic to a product of loops in or . Both and are zero since and are homeomorphic to . Hence every loop in is nullhomotopic.