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Subcompactness And Domain Representability In Go-spaces On Sets Of Real Numbers

机译：实数集上Go-空间中的子紧致性和域可表示性

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In this paper we explore a family of strong completeness properties in GO-spaces defined on sets of real numbers with the usual linear ordering. We show that if r is any GO-topology on the real line R, then (R, τ) is subcompact, and so is any G_δ-subspace of (R, τ). We also show that if (X, τ) is a subcompact GO-space constructed on a subset X c R, then X is a G_δ-subset of any space (R, σ) where σ is any GO-topology on R with τ = σ|x· It follows that, for GO-spaces constructed on sets of real numbers, subcompactness is hereditary to G_δ-subsets. In addition, it follows that if (X, τ) is a subcompact GO-space constructed on any set of real numbers and if τ~s is the topology obtained from τ by isolating all points of a set S is contaned in X, then (X, τ~s) is also subcompact. Whether these two assertions hold for arbitrary subcompact spaces is not known.rnWe use our results on subcompactness to begin the study of other strong completeness properties in GO-spaces constructed on subsets of R. For example, examples show that there are subcompact GO-spaces constructed on subsets X is contaned in R where X is not a G_δ-subset of the usual real line. However, if (X, τ) is a dense-in-itself GO-space constructed on some X is contaned in R and if (X, τ) is subcompact (or more generally domain-representable), then (X, τ) contains a dense subspace Y that is a G_δ-subspace of the usual real line. It follows that (Y, τ │_Y) is a dense subcompact subspace of (X, τ). Furthermore, for a dense-in-itself GO-space constructed on a set of real numbers, the existence of such a dense subspace Y of X is equivalent to pseudo-completeness of (X, τ) (in the sense of Oxtoby). These results eliminate many pathological sets of real numbers as potential counterexamples to the still-open question: "Is there a domain-representable GO-space constructed on a subset of R that is not subcompact"? Finally, we use our subcompactness results to show that any co-compact GO-space constructed on a subset of R must be subcompact.
机译：在本文中，我们探索了在GO空间中由实数集定义的，具有通常线性顺序的一组强完整性属性。我们证明，如果r是实线R上的任何GO拓扑，则（R，τ）是超紧缩的，（R，τ）的任何G_δ-子空间也是如此。我们还表明，如果（X，τ）是在子集X c R上构建的超紧凑GO空间，则X是任何空间（R，σ）的G_δ-子集，其中σ是R上具有τ的R上的任何GO拓扑=σ| x·因此，对于在实数集上构造的GO空间，次紧缩性是G_δ子集的遗传。此外，可以得出结论：如果（X，τ）是在任何实数集上构建的超紧凑型GO空间，并且τ〜s是通过隔离集合S的所有点而从τ获得的拓扑，则包含在X中， （X，τ〜s）也很紧凑。这两个断言是否适用于任意子紧致空间是未知的。rn我们使用子紧致性的结果开始研究在R的子集上构造的GO空间中的其他强完整性属性。例如，示例显示存在子紧致GO空间在子集X上构造的X包含在R中，其中X不是通常实线的G_δ-子集。但是，如果（X，τ）是在某个X上构造的自身稠密的GO空间，包含在R中，并且（X，τ）是超紧凑的（或更一般地域可表示的），则（X，τ）包含一个密集子空间Y，它是通常实线的G_δ-子空间。因此，（Y，τ│_Y）是（X，τ）的密集子紧致子空间。此外，对于在一组实数上构造的自身密集GO空间，X的这种密集子空间Y的存在等效于（X，τ）的伪完全性（在Oxtoby的意义上）。这些结果消除了许多病理实数集，作为仍悬而未决的问题的潜在反例：“是否存在一个在R的子集上构造的，可表示域的GO空间，而该空间不是超紧缩的”？最后，我们使用超紧致度结果表明，在R的子集上构造的任何紧致GO空间都必须是超紧致的。

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