On countable bounded tightness for spaces Cp(X)

It is well known that the space Cp([0, 1]) has countable tightness but it is not Fréchet–Urysohn. Let X be a Cech-complete topological space. We prove that the space Cp(X) of continuous real-valued functions on X endowed with the pointwise topology is Fréchet–Urysohn if and only if Cp(X) has countable bounded tightness, i.e., for every subset A of Cp(X) and every x in the closure of A in Cp(X) there exists a countable and bounding subset of A whose closure contains x. We study also the problem when the weak topology of a locally convex space has countable bounded tightness. Additional results in this direction are provided.  2003 Elsevier Science (USA). All rights reserved.