A note on a theorem of Talagrand

We provide an elementary argument to show that if for a hemicompact k R -space X the space C p ( X ) contains a subset S which separates the points of X and is dominated by irrationals, i.e. S is covered by a family { K α : α ∈ N N } of compact sets such that K α ⊂ K β for α ⩽ β , then C p ( X ) is also dominated by irrationals; consequently C p ( X ) is K-analytic. This fact (which fails for non-hemicompact spaces X) extends an old result of Talagrand.