Scott--Isbell 重合性：超越双完备性的连续 dcpo

中文摘要 AI 辅助

Lawson 和 Mislove 于 1990 年在《拓扑学中的开放问题》中作为问题 535 提出了以下问题：对于核心紧空间 X 和装备其 Scott 拓扑的 dcpo P，在 P 的什么条件下，C(X,P) 上的 Isbell 拓扑和 Scott 拓扑一致？已有证明表明：对于非空双完备连续 dcpo P，C(X,P) 上的 Isbell 拓扑和 Scott 拓扑对每个核心紧空间 X 一致当且仅当 P 是有界完备的；对每个紧核心紧空间 X 一致当且仅当 P 是条件有界完备的；对每个 RW-空间 X 一致当且仅当 P 是带基点的连续 L-域。我们通过结合 Jia、Jung 和 Li 的禁止收缩定理以及关于向下良序链的幂和合适的 Alexandrov 测试空间的分离定理，从所有三个分类中移除了双完备性假设。

英文摘要

Lawson and Mislove posed the following problem in 1990 as Problem~535 in \emph{Open Problems in Topology}: for a core-compact space \(X\) and a dcpo \(P\) equipped with its Scott topology, under what conditions on \(P\) do the Isbell and Scott topologies on \(C(X,P)\) agree?It was proved that, for a nonempty bicomplete continuous dcpo \(P\), the Isbell and Scott topologies on \(C(X,P)\) coincide for every core-compact space \(X\) if and only if \(P\) is bounded complete; for every compact core-compact space \(X\) if and only if \(P\) is conditionally bounded complete; and for every RW-space \(X\) if and only if \(P\) is a pointed continuous \(L\)-domain.We remove the bicompleteness assumption from all three classifications by combining the forbidden-retract theorem of Jia, Jung and Li with a separation theorem for powers of downward well-ordered chains and suitable Alexandrov test spaces.

发表机构

• Sichuan University（四川大学）
• Yancheng Teachers University（盐城师范学院）

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