Resumen de Evaluation maps on products of separable metrizable spaces are Borel

Roman Pol

• Given a completely regular space X we denote by C(X) the space of continuous real - valued functions on X, and let e : X×C(X) ? R, e(x,f) = f(x), be the evaluation map. The function space C(X) is considered with the pointwise topology. We prove the result stated in the title, and give an example of a compact separable space X such that the evaluation map e on X is Borel measurable, but the preimage of some open set under e is not a countable union of closed sets, answering two questions by M.R.Burke, Borel measurability of separately continuous functions II, Top. Appl. 134 (2003), 154 - 188.