Quasicontinuous functions and the topology of uniform convergence on compacta

Let X be a Hausdorff topological space, Q(X,R) be the space of all quasicontinuous functions on X with values in R and ?UC be the topology of uniform convergence on compacta. If X is hemicompact, then (Q(X,R), ?UC) is metrizable and thus many cardinal invariants, including weight, density and cellularity coincide on (Q(X,R), ?UC). We find further conditions on X under which these cardinal invariants coincide on (Q(X,R), ?UC) as well as characterizations of some cardinal invariants of (Q(X,R), ?UC). It is known that the weight of continuous functions (C(R,R), ?UC) is ?0. We will show that the weight of (Q(R,R), ?UC) is 2c.

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 𝒯UC 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 𝒯UC.