Homotopy Characterization of ANR Function Spaces

Homotopy Characterization of ANR Function Spaces

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Let mel axolotl Y be an absolute neighbourhood retract (ANR) for the class of metric spaces and 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 ice blue graphic tee generated Tychonoff space and Y is not discrete, then YX is an ANR for metric spaces if and only if X is hemicompact and YX has the homotopy type of a CW complex.