Quasicontinuous Functions, Densely Continuous Forms and Compactness

Let X be a locally compact space. A subfamily ℱ of the space D*(X, ℝ) of densely continuous forms with nonempty compact values from X to ℝ equipped with the topology 𝒯_UC of uniform convergence on compact sets is compact if and only if {sup(F) : F ∈ ℱ} is compact in the space Q(X, ℝ) of quasicontinuous functions from X to ℝ equipped with the topology 𝒯_UC.