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arXiv · 1001.0596

GO-spaces and Noetherian spectra

David Milovich
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Abstract

The Noetherian type of a space is the least k for which the space has a k^op-like base, i.e., a base in which no element has k-many supersets. We prove some results about Noetherian types of (generalized) ordered spaces and products thereof. For example: the density of a product of not-too-many compact linear orders never exceeds its Noetherian type, with equality possible only for singular Noetherian types; we prove a similar result for products of Lindelof GO-spaces. A countable product of compact linear orders has an omega_1^op-like base if and only if it is metrizable, and every metrizable space has an omega^op-like base. An infinite cardinal k is the Noetherian type of a compact LOTS if and only if k is not omega_1 and k is not weakly inaccessible. There is a Lindelof LOTS with Noetherian type omega_1 and there consistently is a Lindelof LOTS with weakly inaccessible Noetherian type.

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David Milovich. 2010-01-04. GO-spaces and Noetherian spectra. https://arxiv.org/abs/1001.0596

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