The strength of prime separation, sobriety, and compactness theorems
type: Item
instance of: scholarly article
title: The strength of prime separation, sobriety, and compactness theorems
zbMATH Open document ID: 1446.03082
author: Marcel Erné
published in: Topology and its Applications
publication date: 2018-06-25T00:00:00Z
review text: This paper is an important contribution to the study of diverse topological compactness theorems, order-topological dualities, and various topological sobriety concepts in ZF set theory (i.e. Zermelo-Fraenkel set theory without the axiom of choice (AC)). The weak choice principle that has a central role in this paper is the ultrafilter principle (UP) which asserts that every proper set-theoretical filter can be extended to an ultrafilter. [UP is equivalent to the Boolean prime ideal theorem (PIT), which asserts that every non-trivial Boolean algebra has a prime ideal. It is also well-known that PIT (and thus UP) is strictly weaker than AC in ZF.] \par The main body of this paper is divided into four sections. In Section 1 (``Prime separation in distributive lattices and posets''), the separation lemma (SL) (i.e. every distributive complete lattice is $\delta$-separated, that is, any open filter in the lattice is an intersection of completely prime filters), which is equivalent to UP, is thoroughly investigated. In the realm of lattices, and locales in particular (a locale is a complete lattice $L$ such that $x\wedge\bigvee Y=\bigvee\{x\wedge y:y\in Y\}$ for all $x\in L$ and $Y\subseteq L$), an extensive list of equivalents of UP, and hence of SL, is given. This section concludes with the mention of two known important consequences of the principle of dependent choices (DC) concerning continuous posets and supercontinuous lattices, namely ``Continuous posets are strictly continuous'' and ``Supercontinuous lattices are spatial and cospatial'', and the open problem of whether or not DC can be avoided in proving the strict continuity of continuous posets and the spatiality of continuous lattices. Furthermore, the author includes two most informative and detailed diagrams that depict numerous implications in ZFC (= ZF + AC) and in ZF with regard to continuity properties of posets and lattices. \par In Section 2 (``Variants of sobriety''), the author compares the usual co…
reviewed by: Eleftherios Tachtsis
Mathematics Subject Classification ID: 03E25
Mathematics Subject Classification ID: 06D22
Mathematics Subject Classification ID: 54D10
Mathematics Subject Classification ID: 54D30
Mathematics Subject Classification ID: 54D45
zbMATH DE Number: 6894007
zbMATH Keywords: compact
zbMATH Keywords: irreducible
zbMATH Keywords: prime
zbMATH Keywords: (strictly) continuous
zbMATH Keywords: (strictly) sober
zbMATH Keywords: (strictly) spatial
zbMATH Keywords: supercompact
zbMATH Keywords: transversal
zbMATH Keywords: well-filtered
MaRDI profile type: Publication
full work available at URL: j.topol.2018.04.002
OpenAlex ID: W2809211374
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The strength of prime separation, sobriety, and compactness theorems (DOI)
DOI: 10.1016/J.TOPOL.2018.04.002