= Compactly supported cohomology of Euclidean space
{title2=$H_c^q(\mathbb R^d;A)$}
For a nonnegative integer $d$ and an <abelian group> $A$,
$$
H_c^q(\mathbb R^d;A)\cong\begin{cases}A,&q=d,\\0,&q\ne d.\end{cases}
$$
The <one-point compactification> is the <sphere> $S^d$, and it is locally contractible at the added point. Apply the <compact-support comparison with a one-point compactification> and the <reduced cohomology> of a sphere. In dimension zero the compactification is $S^0$, giving the same formula.
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