Quasicontinuous functions, densely continuous forms and compactness
Abstract
Let $X$ be a locally compact space. A subfamily $\mathcal{F}$ of the space
$D^\ast( X ; \mathbb{R} )$ of densely continuous forms with nonempty compact values from $X$
to \mathbb{R} equipped with the topology $\tau_{UC}$ of uniform convergence on compact sets
is compact if and only if ${\rm{sup}(F) : F \in F$ is compact in the space $Q( X; \mathbb{R} )$
of quasicontinuous functions from $X$ to $ \mathbb{R} $ equipped with the topology $\tau_{UC}$.
$D^\ast( X ; \mathbb{R} )$ of densely continuous forms with nonempty compact values from $X$
to \mathbb{R} equipped with the topology $\tau_{UC}$ of uniform convergence on compact sets
is compact if and only if ${\rm{sup}(F) : F \in F$ is compact in the space $Q( X; \mathbb{R} )$
of quasicontinuous functions from $X$ to $ \mathbb{R} $ equipped with the topology $\tau_{UC}$.