  We prove that for generalised partitions of unity ${\phi_i \mid i \in I}$ and
coverings $\mathfrak{U}:={\phi_i^{-1} (R \setminus {0}) \mid i \in I}$ of a
topological space $X$ the cohomology of abstract $\mathfrak{U}$-local cochains
coincides with the cohomology of continuous $\mathfrak{U}$-local cochains,
provided the coefficients are loop contractible. Furthermore we show that for
each locally contractible group $G$ and loop contractible coefficient group $V$
the complex of germs of continuous functions on left-invariant diagonal
neighbourhoods computes the Alexander-Spanier and singular cohomology; similar
results are obtained for $k$-groups and for germs of smooth functions on Lie
groups $G$.
