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Let $X=T^2-\{p\}$ be the torus with one point removed. Since ${p}$ is closed in $T^2$, $X=T^{2}-\{p\}$ is open. In Hausdorff spaces compact subsets are closed, so $X$ is not compact.

I was wondering how to compute the de Rham cohomology with compact supports of $X$.

It's not the same as the standard de Rham cohomology. As $X$ is orientable and connected, $H^2_c(X)=\mathbb R$, but I am struggling with $H^0_c$ and $H^1_c$.

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    $\begingroup$ Worth noting that $T^2 - \{p\}$ is open is only an argument that it is not closed given that $T^2 - \{p\}$ is connected. $\endgroup$ – Keefer Rowan Apr 21 at 22:01
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Using Poincaré duality $$H_c^k \simeq H_{2-k}$$ you can reduce the computation to this answer.

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  • $\begingroup$ Since $M=T^2-\{p\}$ is oriented one could also use the duality $H^k(M)=(H^{n-k}_c(M))^*$. So $H^{2}_c(M)=\mathbb{R}^*$ , $H^{1}_c(M)={\mathbb{R}^{2}}^{*}$ and $H^{0}_c(M)=0^*$. Is that right? $\endgroup$ – NicAG Apr 22 at 17:20
  • $\begingroup$ Yes, this works as well. $\endgroup$ – Paweł Czyż Apr 22 at 20:14

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