On dense subspaces of C-p(X)

For a Tychonoff space X, we denote by C-p(X) the space of all real-valued continuous functions on X with the topology of pointwise convergence. We show the following: (1) if omega(1) is a caliber for every dense subspace of C-p(X), then C-p(X) is (omega(1),omega(1))-narrow; (2) if every dense subspace of C-p(X) is compact-dense in C-p(X), then every non-trivial countable omega-cover of open sets of X contains a gamma-cover. The first result gives the positive answer to Problem 4.4 in [6], and the second one is a partial answer to Problem 4.3 in [6].