I found in this topic or (question) a reason for my question, but i do not understand it. As this question is quite old, I hope someone else can help me.
Assume $U$ is an open subset of a topological space $X$, which is Hausdorff. Then there is an induced map from $H_C^n(U) \to H_C^n X$. Note that I really mean this direction, this map actually really exists. Because I am not so familiar with fiber bundles etc, where you can explicitly define this map, I thought of another way of obtaining this map, like in the answers of the abouve link.
As $U$ is open,the complement $V$ of $U$ is closed. As the inclusion map of a closed subspace is proper, this induces a map on cohomoology with compact support.
$$H_C^n X \to H_C^n V$$. As the inclusion is injective, the induced map on cochain level is surjective, there fore we get a short exact sequence of cochain complexes or a long exact sequence as follows:
$\cdots \to H_C^n (X,V) \to H_C^n X \to H_C^n V \to \cdots $
No thanks to naturality, if I could show that $H_C^n (X,V) $ is isomorphic to $H_C^n U$ I would be happy. Actually I thought of Excision, but I cant really imagine how to do it. As the excision map is an inclusion this might work. I would really appreciate if someone could help me. I am really sure, that this sequence exists. I think the Hausdorff property is used in the Excision argument... If there is anouther reason, I would take it as well. Actually you could work with one-point-compactification, but I did not work this out.