= Nilpotence from a contractible subcomplex cover
{title2=$X=\bigcup_{j=1}^l A_j\Longrightarrow\widetilde h^*(X)^l=0$}
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 $h^*(X,\bigcup_jA_j)=h^*(X,X)=0$, 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.
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