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 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 ( K X ) is perfectly normal and paracompact. However, there is a closed subgroup G of { 0 , 1 } X with X as above whose space of characters G ∧ is not normal.