The strength of prime separation, sobriety, and compactness theorems
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 concept of sobriety for topological spaces with the notion of strict sobriety (or $\delta$-sobriety), and proves -- among several interesting results -- the following: UP is equivalent to each of ``Sober spaces are strictly sober'' and ``Locales isomorphic to topologies of sober spaces are strictly spatial''. [``Sober spaces are strictly sober and well-filtered'' was established by \textit{K. H. Hofmann} and \textit{M. W. Mislove} [Lect. Notes Math. 871, 209--248 (1981; Zbl 0464.06005)] in the setting of ZFC.] An open problem in this area is whether in $\mathbf{ZF}$ arbitrary Hausdorff spaces are strictly sober (whereas, in ZF, Hausdorff spaces are sober and well-filtered). \par In Section 3 (``Compactness, irreducibility, and transversals''), the author establishes the equivalence of UP to several basic topological principles concerning certain filter bases of compact sets. For example, one of these principles is the irreducible transversal theorem which states that any collection $\mathcal{K}$ of compact sets whose saturations form a filter base has an irreducible transversal (a transversal of $\mathcal{K}$ is a subset of $\bigcup\mathcal{K}$ which meets all members of $\mathcal{K}$, and the saturation of a set is the intersection of all its neighborhoods). \par In Section 4 (``Categorical dualities for sober spaces''), the author shows that central theorems in the theory of algebraic or continuous frames, which concern equivalences and dualities between categories of ordered structures and topological spaces, are equivalent to UP. Among others, the Stone duality between bounded distributive lattices and spectral spaces, and the Stone duality between Boolean algebras and Boolean spaces. \par Concluding this review, we would like to mention the notable fact that the author generously includes a ``Notes'' part at the end of each section providing the reader with important details of the history and the advance of research in this area, as well as with a concise analysis of the proven material and its connection with related works of other researchers. The author's commendable work concludes with a list of 63 bibliographical entries.
scientific article; zbMATH DE number 6894007
scientific article