Every open covering of a "Ck" manifold "M" has a "Ck" partition of unity. This allows for certain constructions from the topology of "Ck" functions on R"n" to be carried over to the category of differentiable manifolds. In particular, it is possible to discuss integration by choosing a partition of unity subordinate to a particular coordinate atlas, and carrying out the integration in each chart of R"n". Partitions of unity therefore allow for certain other kinds of function spaces to be considered: for instance L"p" spaces, Sobolev spaces, and other kinds of spaces that require integration.