Domain-完备与LCS-完备空间
Domain-complete and LCS-complete spaces
摘要：
我们研究连续dcpos的$G_δ$子空间，我们称之为域完备空间，和局部紧sober空间的$G_δ$子空间，我们称之为LCS完备空间。这些空间包括所有局部紧的sober空间--特别是所有连续的dcpos空间--，所有Čech意义下的拓扑完备空间，以及所有拟波兰空间--特别是所有波兰空间。我们证明了LCS-完备空间是sober、Wilker、紧Choquet-完备、完全Baire和$\odot$ -辅音-特别是辅音空间;域完备）空间是准波兰空间;并且可度量的LCS完备（分别，域完备空间是完全可度量化的空间。我们包括两个应用程序：在LCS-完备空间上，所有的连续赋值都扩张到测度上，并且次线性预影形成了一个与连续赋值空间的凸Hoare幂域同胚的空间。
英文摘要：
We study $G_δ$ subspaces of continuous dcpos, which we call domain-complete spaces, and $G_δ$ subspaces of locally compact sober spaces, which we call LCS-complete spaces. Those include all locally compact sober spaces-in particular, all continuous dcpos-, all topologically complete spaces in the sense of Čech, and all quasi-Polish spaces-in particular, all Polish spaces. We show that LCS-complete spaces are sober, Wilker, compactly Choquet-complete, completely Baire, and $\odot$-consonant-in particular, consonant; that the countably-based LCS-complete (resp., domain-complete) spaces are the quasi-Polish spaces exactly; and that the metrizable LCS-complete (resp., domain-complete) spaces are the completely metrizable spaces. We include two applications: on LCS-complete spaces, all continuous valuations extend to measures, and sublinear previsions form a space homeomorphic to the convex Hoare powerdomain of the space of continuous valuations.
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