On Noetherian Spaces

J. Goubault-Larrecq
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引用次数: 37

Abstract

A topological space is Noetherian iff every open is compact. Our starting point is that this notion generalizes that of well-quasi order, in the sense that an Alexandroff-discrete space is Noetherian iff its specialization quasi-ordering is well. For more general spaces, this opens the way to verifying infinite transition systems based on non-well quasi ordered sets, but where the preimage operator satisfies an additional continuity assumption. The technical development rests heavily on techniques arising from topology and domain theory, including sobriety and the de Groot dual of a stably compact space. We show that the category Nthr of Noetherian spaces is finitely complete and finitely cocomplete. Finally, we note that if X is a Noetherian space, then the set of all (even infinite) subsets of X is again Noetherian, a result that fails for well-quasi orders.
关于诺瑟空间
如果每个开都是紧的,那么拓扑空间就是诺瑟空间。我们的出发点是这个概念推广了井-拟序的概念,从某种意义上说,如果亚历山德罗夫离散空间的专门化拟序很好,那么它就是诺瑟空间。对于更一般的空间,这为验证基于非井拟有序集的无限过渡系统开辟了道路,但其中的预像算子满足附加的连续性假设。技术的发展很大程度上依赖于拓扑学和领域理论,包括稳定紧空间的清醒性和德格鲁特对偶。证明了noether空间的范畴Nthr是有限完备和有限协完备的。最后,我们注意到如果X是一个诺埃尔空间,那么X的所有(甚至无限)子集的集合仍然是诺埃尔的,这个结果对于井-拟序是不成立的。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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