Completeness properties of the space of quasicontinuous functions
arXiv:2608.12318
Abstract
Quasicontinuous functions have found applications in many areas of mathematics. We study completeness properties of the space of quasicontinuous functions equipped with the topology of pointwise convergence. Let X be a Hausdorff topological space, Q(X) be the space of quasicontinuous real-valued functions and τ_p be the topology of the pointwise convergence. For (Q(X), τ_p) complete metrizability, Polishness and Cech-completeness are equivalent. If (Q(X), τ_p) is completely metrizable, then X is countable and the set I(X) of isolated points of X is dense in X. If X is first countable, then (Q(X), τ_p) is completely metrizable if and only if X is countable and I(X) is dense in X.
11 pages