For $Z\subseteq A\subseteq X$ with $\cl_X(Z)\subseteq A^\circ$, removing $Z$ from $X$ and $A$ does not change $H_*(X, A)$.Suggest a slogan
Given subspaces \(Z \subseteq A \subseteq X\) with \(\cl_X(Z) \subseteq \interiorof{A}\), the inclusion of pairs induces isomorphisms \[ H_n(X\sm Z,\, A\sm Z) \mapsvia{\sim} H_n(X, A) \qquad \text{for all } n .\] Equivalently, for subspaces \(A, B\subseteq X\) with \(X = \interiorof{A} \union \interiorof{B}\), the inclusion \((B, A\intersect B)\injects (X, A)\) induces isomorphisms \(H_n(B, A\intersect B)\mapsvia{\sim} H_n(X,A)\) for all \(n\). The two forms are related by \(B = X\sm Z\) [@Hat02].