Homotopy Characterization of ANR Function Spaces

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