Fragmentability and compactness in C(K)-spaces

Let K be a compact Hausdorff space, ${\displaystyle C_p\left(K\right)}$ the space of continuous functions on K endowed with the pointwise convergence topology, D ⊂ K a dense subset and ${\displaystyle t_p\left(D\right)}$ the topology in C(K) of pointwise convergence on D. It is proved that when ${\displaystyle C_p\left(K\right)}$ is Lindelöf the ${\displaystyle t_p\left(D\right)}$-compact subsets of C(K) are fragmented by the supremum norm of C(K). As a consequence we obtain some Namioka type results and apply them to prove that if K is separable and ${\displaystyle C_p\left(K\right)}$ is Lindelöf, then K is metrizable if, and only if, there is a countable and dense subset D ⊂ K such that ${\displaystyle (C\left(K\right),t_p\left(D\right))}$ is analytic. We also show that if K is a separable Rosenthal compact space, then K is metrizable if, and only if, ${\displaystyle C_p\left(K\right)}$ is Lindelöf. We complete our study by showing that if K does not contain a copy of βℕ, then convex ${\displaystyle t_p\left(D\right)}$-compact subsets of C(K) have the weak Radon-Nikodym property.