空间的强完备化
Strong completions of spaces
摘要：
一个拓扑空间的非空子集是不可约的，如果每当它被两个闭集的并覆盖时，它就已经被其中一个覆盖了。不可约集在增殖中出现：（1）每一个单点集都是不可约的，（2）偏序集的有向子集（在Domain理论中具有基础地位）恰好是它的Alexandroff不可约集，（3）$T_0$空间的有向子集（关于特化阶）总是不可约的，（4）每一个不可约集的拓扑闭包也是不可约的。近年来，不可约集在Domain理论和非Hausdorff拓扑中的应用得到了扩展。值得注意的是，Zhao和Ho（2009）通过选择不可约集作为有向集的拓扑替代，直接在$T_0$空间的背景下发展了Domain理论的核心。正如有向子集的上确界的存在性在Domain理论中很突出（因此也有dcpo的概念-所有有向上确界都存在的偏序集），赵和何（2009）发展的拓扑Domain理论中的不可约子集也是如此。dcpo的拓扑对应是这样的：一个$T_0$空间被称为强完备的，如果所有不可约子集的上确界存在。本文证明了强完备$T_0$空间形式的范畴$\mathbf{scTop^+}$是$T_0$空间的某个lluf子范畴$\mathbf{Top^+}$的反射子范畴.
英文摘要：
A non-empty subset of a topological space is irreducible if whenever it is covered by the union of two closed sets, then already it is covered by one of them. Irreducible sets occur in proliferation: (1) every singleton set is irreducible, (2) directed subsets (which of fundamental status in domain theory) of a poset are exactly its Alexandroff irreducible sets, (3) directed subsets (with respect to the specialization order) of a $T_0$ space are always irreducible, and (4) the topological closure of every irreducible set is again irreducible. In recent years, the usefulness of irreducible sets in domain theory and non-Hausdorff topology has expanded. Notably, Zhao and Ho (2009) developed the core of domain theory directly in the context of $T_0$ spaces by choosing the irreducible sets as the topological substitute for directed sets. Just as the existence of suprema of directed subsets is featured prominently in domain theory (and hence the notion of a dcpo -- a poset in which all directed suprema exist), so too is that of irreducible subsets in the topological domain theory developed by Zhao and Ho (2009). The topological counterpart of a dcpo is thus this: A $T_0$ space is said to be strongly complete if the suprema of all irreducible subsets exist. In this paper, we show that the category, $\mathbf{scTop^+}$, of strongly complete $T_0$ spaces forms are reflective subcategory of a certain lluf subcategory, $\mathbf{Top^+}$, of $T_0$ spaces.
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