Cohomology with compact support

In mathematics, cohomology with compact support refers to certain cohomology theories, usually with some condition requiring that cocycles should have compact support.

Let ${\displaystyle X}$ be a topological space. Then

This is also naturally isomorphic to the cohomology of the sub–chain complex ${\displaystyle C_{c}^{\ast }(X;R)}$ consisting of all singular cochains ${\displaystyle \phi :C^{i}(X;R)\to R}$ that have compact support in the sense that there exists some compact ${\displaystyle K\subseteq X}$ such that ${\displaystyle \phi }$ vanishes on all chains in ${\displaystyle X\setminus K}$.

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