Nilpotence from a contractible subcomplex cover ($X={\bigcup }_{j=1}^l{A}_j⟹{\tilde{h}}^{\ast }(X{)}^l=0$)

If a pointed CW complex is covered by $l$ contractible subcomplexes containing the basepoint, every reduced class lifts to a relative class for each appropriate subcomplex. Their relative cup product lies in $h^{\ast }(X,{\bigcup }_j{A}_j)=h^{\ast }(X,X)=0$, so every product of $l$ reduced classes vanishes. The conclusion also holds if $X$ is path connected. Without either condition a two-point space covered by its two points is a counterexample in ordinary degree-zero cohomology.