Distinguished dense C-subspaces

Let C p ( X ) be the linear space of real-valued continuous functions with the pointwise topology. It is known that a Tychonoff space X is a Δ-space if and only if the locally convex space C p ( X ) is distinguished. It has been recently shown that if there is a continuous linear surjection from C p ( X ) onto C p ( Y ) and X is a Δ-space, Y is also a Δ-space. Here we investigate under what conditions the presence of a dense distinguished subspace E in C p ( X ) leads X to be a Δ-space. We also produce a class of spaces X ∉ Δ for which C p ( X ) contains a distinguished dense subspace.