拟连续函数空间的完备性性质
Completeness properties of the space of quasicontinuous functions
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中文总结 AI 辅助
本文研究配备逐点收敛拓扑的拟连续函数空间的完备性性质，明确其完全可度量化、波兰性与Čech完备性等价，并给出该空间完全可度量化的充要条件。
中文摘要 AI 辅助
拟连续函数已在数学诸多领域得到应用。本文研究配备逐点收敛拓扑的拟连续函数空间的完备性性质。设X为Hausdorff拓扑空间，Q(X)为取值于实数域的拟连续函数空间，τₚ为逐点收敛拓扑。对于(Q(X), τₚ)，其完全可度量化、波兰性与Čech完备性三者等价；若(Q(X), τₚ)完全可度量化，则X必为可数集且X的孤立点集I(X)在X中稠密；若X为第一可数空间，则(Q(X), τₚ)完全可度量化当且仅当X为可数集且I(X)在X中稠密。
英文摘要
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.
