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 KXKX 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(KX) 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∧ is not normal.