Homotopy characterization of ANR function spaces

Let y be an absolute neighbourhood retract (anr) for the class of metric spaces and let x be a topological space. let y x 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 y x is an anr for metric spaces if and only if x is hemicompact and y x has the homotopy type of a cw complex. © 2013 jaka smrekar.