= Collapsing a pair theorem
For a <good pair> $(X,A)$ with $A\ne\varnothing$, the quotient map induces natural <isomorphisms>
$$
H_q(X,A)\cong\widetilde H_q(X/A),\qquad q\geq0.
$$
Here $X/A$ collapses the entire subspace $A$ to one <base point>. A neighborhood <deformation retraction> and <excision> compare the relative <homology> with the quotient, so computing a pair becomes computing the <reduced homology> of one space.
If $A=\varnothing$, use the based convention $X/\varnothing=X_+$, adjoining a disjoint <base point>, to retain the formula. With the unbased quotient convention $X/\varnothing=X$, the formula would fail in degree zero. For the <closed disk> and its boundary <sphere>, the quotient is $S^n$ and recovers the <relative homology of a simplex and its boundary>.
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