Homotopy Characterization of ANR Function Spaces

Let be an absolute neighbourhood retract (ANR) for the class of metric spaces and let be a topological space. Let denote the space of continuous maps from to equipped with the compact open topology. We show that if is a compactly generated Tychonoff space and is not discrete, then is an ANR for metric spaces if and only if is hemicompact and has the homotopy type of a CW complex.