The existing topological representation of an orthocomplemented lattice via the clopen orthoregular subsets of a Stone space depends upon Alexander’s Subbase Theorem, which asserts that a topological ...space
X
is compact if every subbasic open cover of
X
admits of a finite subcover. This is an easy consequence of the Ultrafilter Theorem—whose proof depends upon Zorn’s Lemma, which is well known to be equivalent to the Axiom of Choice. Within this work, we give a choice-free topological representation of orthocomplemented lattices by means of a special subclass of spectral spaces; choice-free in the sense that our representation avoids use of Alexander’s Subbase Theorem, along with its associated nonconstructive choice principles. We then introduce a new subclass of spectral spaces which we call
upper Vietoris orthospaces
in order to characterize up to homeomorphism (and isomorphism with respect to their orthospace reducts) the spectral spaces of proper lattice filters used in our representation. It is then shown how our constructions give rise to a choice-free dual equivalence of categories between the category of orthocomplemented lattices and the dual category of upper Vietoris orthospaces. Our duality combines Bezhanishvili and Holliday’s choice-free spectral space approach to Stone duality for Boolean algebras with Goldblatt and Bimbó’s choice-dependent orthospace approach to Stone duality for orthocomplemented lattices.
ITERATING SYMMETRIC EXTENSIONS KARAGILA, ASAF
The Journal of symbolic logic,
03/2019, Letnik:
84, Številka:
1
Journal Article
Recenzirano
Odprti dostop
The notion of a symmetric extension extends the usual notion of forcing by identifying a particular class of names which forms an intermediate model of ZF between the ground model and the generic ...extension, and often the axiom of choice fails in these models. Symmetric extensions are generally used to prove choiceless consistency results. We develop a framework for iterating symmetric extensions in order to construct new models of ZF. We show how to obtain some well-known and lesser-known results using this framework. Specifically, we discuss Kinna–Wagner principles and obtain some results related to their failure.
Ramsey’s Theorem is naturally connected to the statement “every infinite partially ordered set has either an infinite chain or an infinite anti-chain”. Indeed, it is a well-known result that Ramsey’s ...Theorem implies the latter principle. In the book “Consequences of the Axiom of Choice” by P. Howard and J. E. Rubin, it is stated as unknown whether the above implication is reversible, that is whether the principle “every infinite partially ordered set has either an infinite chain or an infinite anti-chain” implies Ramsey’s Theorem. The purpose of this paper is to settle the aforementioned open problem. In particular, we construct a suitable Fraenkel–Mostowski permutation model ${\cal N}$ for ZFA and prove that the above principle for infinite partially ordered sets is true in ${\cal N}$, whereas Ramsey’s Theorem is false in ${\cal N}$. Then, based on the existence of ${\cal N}$ and on results of D. Pincus, we show that there is a model of ZF which satisfies “every infinite partially ordered set has either an infinite chain or an infinite anti-chain” and the negation of Ramsey’s Theorem. In addition, we prove that Ramsey’s Theorem (hence, the above principle for infinite partially ordered sets) is true in Mostowski’s linearly ordered model, filling the gap of information in the book “Consequences of the Axiom of Choice”.
We prove that it is relatively consistent with \mathsf {ZF} (i.e., Zermelo-Fraenkel set theory without the Axiom of Choice ( \mathsf {AC})) that the Axiom of Countable Choice ( \mathsf {AC}^{\aleph ..._{0}}) is true, but the Urysohn Lemma ( \mathsf {UL}), and hence the Tietze Extension Theorem ( \mathsf {TET}), is false. This settles the corresponding open problem in P. Howard and J. E. Rubin Consequences of the Axiom of Choice , Mathematical Surveys and Monographs, Vol. 59, American Mathematical Society, Providence, RI, 1998. We also prove that in Läuchli's permutation model of \mathsf {ZFA} + \neg \mathsf {UL}, \mathsf {AC}^{\aleph _{0}} is false. This fills the gap in information in the above monograph of Howard and Rubin.
Hausdorff compactifications in ZF Keremedis, Kyriakos; Wajch, Eliza
Topology and its applications,
05/2019, Letnik:
258
Journal Article
Recenzirano
Odprti dostop
For a compactification αX of a Tychonoff space X, the algebra of all functions f∈C(X) that are continuously extendable over αX is denoted by Cα(X). It is shown that, in a model of ZF, it may happen ...that a discrete space X can have non-equivalent Hausdorff compactifications αX and γX such that Cα(X)=Cγ(X). Amorphous sets are applied to a proof that Glicksberg's theorem that βX×βY is the Čech-Stone compactification of X×Y when X×Y is a Tychonoff pseudocompact space is false in some models of ZF. It is noticed that if all Tychonoff compactifications of locally compact spaces had C⁎-embedded remainders, then van Douwen's choice principle would be satisfied. Necessary and sufficient conditions for a set of continuous bounded real functions on a Tychonoff space X to generate a compactification of X are given in ZF. A concept of a maximal functional compactification is investigated in the absence of the axiom of choice.
On Iso-dense and Scattered Spaces without AC Keremedis, Kyriakos; Tachtsis, Eleftherios; Wajch, Eliza
Resultate der Mathematik,
08/2023, Letnik:
78, Številka:
4
Journal Article
Recenzirano
Odprti dostop
A topological space is iso-dense if it has a dense set of isolated points, and it is scattered if each of its non-empty subspaces has an isolated point. In
ZF
(i.e. Zermelo–Fraenkel set theory ...without the Axiom of Choice (
AC
)), basic properties of iso-dense spaces are investigated. A new permutation model is constructed, in which there exists a discrete weakly Dedekind-finite space having the Cantor set as a remainder; the result is transferable to
ZF
. This settles an open problem posed by Keremedis, Tachtsis and Wajch in 2021. A metrization theorem for a class of quasi-metric spaces is deduced. The statement “Every compact scattered metrizable space is separable” and several other statements about metric iso-dense spaces are shown to be equivalent to the axiom of countable choice for families of finite sets. Results related to the open problem of the set-theoretic strength of the statement “Every non-discrete compact metrizable space contains an infinite compact scattered subspace” are also included.
Choiceless chain conditions Karagila, Asaf; Schweber, Noah
European journal of mathematics,
11/2022, Letnik:
8, Številka:
Suppl 2
Journal Article
Odprti dostop
Chain conditions are one of the major tools used in the theory of forcing. We say that a partial order has the countable chain condition if every antichain (in the sense of forcing) is countable. ...Without the axiom of choice antichains tend to be of little use, for various reasons, and in this short note we study a number of conditions which in
ZFC
are equivalent to the countable chain condition.
We show that Countable-to-one Uniformization is preserved by forcing with P(ω)/Fin over a model of ZF in which every set of reals is completely Ramsey. We also give an exposition of Todorcevic's ...theorem that Ramsey ultrafilters are generic for P(ω)/Fin over suitable inner models.
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