= Cap-product support lemma for a two-set cover
A valid support statement for the <cap product> uses a relative class. Let $M$ be a <closed manifold>[closed], oriented $n$-dimensional <smooth manifold>, and let $W\subset M$ be a compact codimension-zero <submanifold> with boundary. If $\alpha\in H^k(M;R)$ restricts to zero on $M\setminus\operatorname{int}W$, then
$$
[M]\frown\alpha\ \in\ \operatorname{im}\bigl(H_{n-k}(W;R)\to H_{n-k}(M;R)\bigr).
$$
Indeed, the pair <long exact sequence in cohomology> lifts $\alpha$ to $H^k(M,M\setminus\operatorname{int}W;R)$. <Excision>, using a boundary collar, identifies this with $H^k(W,\partial W;R)$. <Poincare-Lefschetz duality> gives a class in $H_{n-k}(W;R)$ whose image is the desired <cap product>.
Vanishing restrictions on both sets of an open cover do not force this <cap product> to vanish. Cover a <circle> by two proper open arcs. Its generator $\alpha\in H^1(S^1;\mathbb Z)$ restricts to zero on each arc, but $[S^1]\frown\alpha$ generates $H_0(S^1;\mathbb Z)$ and is nonzero. The valid <vanishing cup product from an open cover> instead says that $\alpha\smile\beta=0$ when $\alpha$ vanishes on one covering set and $\beta$ on the other.
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