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arXiv · 2608.12318

Completeness properties of the space of quasicontinuous functions

Ľubica Holá
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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 {\tau}_p be the topology of the pointwise convergence. For (Q(X), {\tau}_p) complete metrizability, Polishness and Cech-completeness are equivalent. If (Q(X), {\tau}_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), {\tau}_p) is completely metrizable if and only if X is countable and I(X) is dense in X.

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Ľubica Holá. 2026-08-12. Completeness properties of the space of quasicontinuous functions. https://arxiv.org/abs/2608.12318

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