Why is $H_c^0(X; G) = 0$ for an arbitrary path-connected and noncompact space $X$?

Question

This is a Hatcher problem, 3.3.20. If $X$ is a manifold, then we have $H_c^0(X; R) \simeq H_n(X; R) \simeq 0$ by Poincare Duality and Hatcher Prop 3.29. In slightly more generality, if $X$ is locally compact and Hausdorff, then we can potentially look at the one-point compactification of $X$ and derive something from that. But I'm really stumped by the arbitrary topological space. How should I approach this problem?

Answer 1

You can just compute this directly from the definition. Let $\alpha$ be any compactly supported $0$-cocycle. For any $x\in X$, choose a path $\gamma$ from $x$ to a point $y$ which is not in the support of $\alpha$ (this is possible since $X$ is path-connected and the support of $\alpha$ is not all of $X$). Then $$0=\delta\alpha(\gamma)=\alpha(\partial\gamma)=\alpha(y)-\alpha(x)=-\alpha(x).$$ Since $x\in X$ was arbitrary, this means $\alpha=0$.

More conceptually, a $0$-cocycle is exactly a function on $X$ which is constant on path-components. So, if $X$ is path-connected, any $0$-cocycle must be constant on all of $X$, and so its support must be all of $X$ unless it is $0$.