Note on K-analyticity and normality in function spaces

We prove that whenever X is zero-dimensional metrizable with σ-compact set of accumulation points and K   is compact metrizable, the function space K^X$K^X$ endowed with the compact-open topology is a compact-covering image of the product of the irrationals and the Cantor cube. In particular, for any metrizable E  , the iterated function space E^(KX)$E^{(K^X)}$ is perfectly normal and paracompact. However, there is a closed subgroup G   of {0,1}^X${\{0,1\}}^X$ with X   as above whose space of characters G^∧$G^{\wedge }$ is not normal.