Spaces of functions and sections with paracompact domain

We study spaces of continuous functions and sections with domain a paracompact Hausdorff k-space $X$ and range a nilpotent CW complex $Y$ , with emphasis on localization at a set of primes. For $\mathop {\rm map}\nolimits _\phi (X,\,Y)$ , the space of maps with prescribed restriction $\phi$ on a suitable subspace $A\subset X$ , we construct a natural spectral sequence of groups that converges to $\pi _*(\mathop {\rm map}\nolimits _\phi (X,\,Y))$ and allows for detection of localization on the level of $E^2$ . Our applications extend and unify the previously known results.