Compact open topology and CW homotopy type

The class W of spaces having the homotopy type of a CW complex is not closed under formation of function spaces. In 1959, Milnor proved the fundamental theorem that, given a space Y∈W and a compact Hausdorff space X, the space YX of continuous functions X→Y, endowed with the compact open topology, belongs to W. P.J. Kahn extended this in 1982, showing that YX∈W if X has finite n-skeleton and πk(Y)=0, k>n. a different approach, we obtain a further generalization and give interesting examples of function spaces YX∈W where X∈W is not homotopy equivalent to a finite complex, and Y∈W has infinitely many nontrivial homotopy groups. We also obtain information about some topological properties that are intimately related to CW homotopy type. application we solve a related problem concerning towers of fibrations between spaces of CW homotopy type.