# “Cohomology at the infinity”: what does one call it

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.

This is the cohomology of $X$ at $\infty$. You'll find a discussion of (singular) homology and cohomology at $\infty$ in Hughes & Ranicki, Ends of Complexes (CUP, 1996).