The limit space Cc(X,Y) and the connected components of X and Y

Let X and Y be limit spaces (in the sense of FISCHER). For f ? C(X, Y), let [f] denote the subset of C(X, Y), where the maps take the connected components of X into those of Y quite analogously to f. The subspace [f] of the continuous convergence space C_c(X, Y) is written as a product II C_c(X_i, Y_k(i)), where X_i runs through the components of X and Y_k(i) always is the component of Y which contains the set f(X_i). Sufficient conditions for the representation C_c(X, Y) = Σ [f] are given (in terms of the spaces X and Y). Some applications on limit homeomorphism groups are included.