Completeness properties of the space of quasicontinuous functions
Ľubica Holá
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 τp be the topology of the pointwise convergence. For (Q(X), τp) complete metrizability, Polishness and Cech-completeness are equivalent. If (Q(X), τp) 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), τp) is completely metrizable if and only if X is countable and I(X) is dense in X.
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