A Study of Function Spaces through a Functor

Let X be a locally compact Hausdorff space and let Fn (X) = limit of the function spaces of maps of X into certain spaces of type K (, n) each of the spaces of sequences SPn, WSPn+1, WSPn+2,. . . , WSP n+m,. . . is a space of type K (, n), SPn WSPn+1 W2SPn+2. . . Wm SP n+m Wm+1SPn+m+1. . . . For any space X, we define the space Fn, m (X) = (Wm SPn+m) X topologized by the compact-open topology. The aim of this paper is i) to investigate the properties of Fn, m (X), ii) to study of the object Fn, m.