QUASICONTINUOUS FUNCTIONS AND THE TOPOLOGY OF UNIFORM CONVERGENCE ON COMPACTA

Abstract

Let $X$ be a Hausdorff topological space, $Q(X,\Bbb R)$ be the space of all quasicontinuous functions on $X$  with values in $\Bbb R$ and $\tau_{UC}$ be the topology of uniform convergence on compacta. If $X$ is hemicompact, then $(Q(X,\Bbb R),\tau_{UC})$ is metrizable and thus many cardinal invariants, including weight, density and cellularity coincide on $(Q(X,\Bbb R),\tau_{UC})$. We find  further conditions on $X$ under which these cardinal invariants coincide on $(Q(X,\Bbb R),\tau_{UC})$ as well as  characterizations of some cardinal invariants of $(Q(X,\Bbb R),\tau_{UC})$. It is known that the weight of continuous functions $(C(\Bbb R,\Bbb R),\tau_{UC})$ is $\aleph_0$. We will show that the  weight of $(Q(\Bbb R,\Bbb R),\tau_{UC})$ is $2^c$.