= Compact-support comparison with a one-point compactification
{title2=$H_c^*(X;A)\cong\widetilde H^*(X^+;A)$}
If the <one-point compactification> $X^+$ is Hausdorff and has a basis of contractible neighbourhoods at the added point, then
$$
H_c^*(X;A)\cong\widetilde H^*(X^+;A).
$$
The <Excision theorem> identifies $H^*(X,X\setminus K)$ with $H^*(X^+,X^+\setminus K)$. The contractible neighbourhoods form a cofinal family of the complements of compact $K$, and their pair <long exact sequences> identify each relative group naturally with <reduced cohomology>. Passing to the <direct limit> gives the comparison. Local contractibility matters for <singular cohomology>; the <Hawaiian earring> illustrates its failure.
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