Suppose $X$ is a "good enough" Hausdorff topological space; we assume that $X$ is *not* compact. Now, for a natural number $k$ and an abelian group $G$, consider the group $\varinjlim_{\substack{U\subseteq X\\ \text{$X\setminus U$ is compact}}} H^k(U,G)$.

Could you, please, give me a reference to a text where this object is defined? I would like to learn the standard term and notation for it.

Thank you in advance,

Serge