If $M$ is a connected manifold,

$$H^0(M)\simeq\mathbb{R}\qquad H^0_c(M) \begin{cases} \simeq\mathbb{R} & \text{if $M$ is compact}\\ =0 & \text{if $M$ is not compact} \end{cases}$$

$$H^0(M)\simeq\mathbb{R}\qquad H^0_c(M) \begin{cases} \simeq\mathbb{R} & \text{if $M$ is compact}\\ =0 & \text{if $M$ is not compact} \end{cases}$$

As in the previous examples, $\text{im }\mathrm{d}_{-1}=0$ and $\text{ker }\mathrm{d}_0$ is made up by constant functions and is isomorph to $\mathbb{R}$, except for the compact cohomology with $M$ non-compact, in which $\text{ker }\mathrm{d}_0$ is also 0

This matches exactly the fact that $H_0(X)\simeq\mathbb{Z}$ (or $\mathbb{R}$, depending on the ring used) for $X$ a path-connected topological space. That is, (co-)homology of order zero represents connected components

This matches exactly the fact that $H_0(X)\simeq\mathbb{Z}$ (or $\mathbb{R}$, depending on the ring used) for $X$ a path-connected topological space. That is, (co-)homology of order zero represents connected components