DANGER !

In the following papers, we work in set theory without the axiom of choice, ZF.

Commutative rings

Lexicographic orders

Thesis

Topology of linearly ordered sets

Measure and integration

General topology

Theoretical Informatics

Hahn-Banach and the Axiom of Choice

Note de synthèse

James sequences and James's sup Theorem

Weak compactness and Axiom of Choice

• Axiome du Choix dénombrable et compacité faible. , Exposé au séminaire ERMIT, Februar 2007.
• Countable choice and compactness , Topology and Applications, Volume 155, Issue 10, (2008) Pages 1077-1088.

• Uniform Eberlein spaces and the finite Axiom of Choice or arXiv preprint.
• Compacité faible dans l^2(I) et axiome du choix (séminaire ERMIT, mai 2008) .
• Uniform Eberlein Compactness and the Axiom of Choice, Abstract, Set Theory, Topology and Banach Spaces (Second International Topology Conference in Kielce, Poland) July 7-11, 2008.
• Notions of Compactness for special subsets of IR^I and some Weak Forms of the Axiom of Choice , Journal of Symbolic Logic , Volume 75, Issue 1 (2010), 255-268.

Rado's selection lemma

• [Rado2010] Rado's selection lemma implies Hahn-Banach, Abstract, 25th Summer Conference on Topology and its Applications, July 25-30, 2010, Kielce Poland.
• [Rado2012] Some consequences of Rado's selection lemma, Archive for Mathematical Logic, vol.51, no7, p.739-749, 2012.

The continuous Hahn-Banach property

• The continuous Hahn-Banach property and the Axiom of Choice , talk at the ICBSOS conference (Nankai University, Tianjin, China), July 2007.
• Uniform Gâteaux differentiability yields Hahn-Banach, Quaestiones Mathematicae, Volume 33, Issue 2, 2010, Pages 131-146.

• Compactness of generalized Helly spaces , Abstract, talk at the 24th Summer Conference on Topology and Its Applications, (Brno, Czech Republic), July 2009.
• Helly spaces and Radon measures on complete lines, Order, vol.29, no3, p.419-441,2012. download.
  We show in ZF that for every complete line L (thus L endowed with the order topology is a compact Hausdorff space), the Banach space C(L) satisfies the (multiple) continuous Hahn-Banach property, and that the dual ball of C(L) is compact in the weak* topology.

• A remark on the continuous Hahn-Banach property., 26th Summer Conference on Topology and Its Applications, July 26-29, 2011, City College of CUNY New York, NY, USA.
• Three-space type Hahn-Banach properties,, Mathematical Logic Quarterly, vol.63, no5, p.320-333 (2017).
  We show in ZF that the continuous Hahn-Banach property is a ``three-space type property''. We deduce that for every scattered compact Hausdorff topological space K, the Banach space C(K) satisfies the continuous Hahn-Banach property. We also prove in ZF Rudin's theorem: ``Every Radon measure on a scattered compact Hausdorff topological space is discrete.'' The following question stays open: "Given a compact Hausdorff topological space K, does the Banach space C(K) satisfy the continuous Hahn-Banach property in ZF?''

Discrete mathematics

• with C.Delhommé , Spanning Trees and the Axiom of Choice, Reports on Mathematical Logic, 2006. download.
  
• with P.Spinelli, Nash equilibria and values through modular partitions in infinite games. Discrete Mathematics 312(6): 1201-1212 (2012). download.

Existence of linear forms and Axiom of Choice

• Formes linéaires et axiomes de choix fini , Exposé au séminaire ERMIT, février 2008.
• Puissances réduites et existence de formes linéaires , Exposé au séminaire ERMIT, septembre 2008.
• Linear forms and axioms of choice, Commentationes Mathematicae Universitatis Carolinae, 50,3 (2009) 421-431.
• Linear extenders and the Axiom of Choice, Commentationes Mathematicae Universitatis Carolinae, vol.58 no4, p.419-434 (2017).
  We consider for every commutative field IK the statement D(IK): ``Every non null IK-vector space has a non null linear form.'' The statement D(IK) is not provable in ZF. We show that for every field IK, the statement D(IK) implies the following stronger statement: ``Every vector subspace of a IK-vector space has a linear extender.'' Here, given a vector subspace F of a vector space E, a ``linear extender'' on F is a linear mapping associating to every linear form f on F, a linear mapping g on E extending f. We also introduce for each spherically complete valued field (IK,|.|) a ``Hahn-Banach''-type statement D(IK,|.|) (Ingleton's statement) and we prove that D(IK,|.|) is equivalent to the existence of isometric linear extenders for vector subspaces of ultrametric normed spaces over the valued field (IK,|.|).
• Multiple Choices imply the Ingleton and Krein-Milman axioms, Journal of Symbolic Logic (2020). In set theory without the Axiom of Choice, we consider Ingleton's axiom which is the counterpart in ultrametric analysis of the Hahn-Banach axiom. We show that in ZFA, set theory without the Axiom of Choice weakened to allow "atoms", Ingleton's axiom does not imply the Axiom of Choice: this solves in ZFA a question raised by van Rooij (1992). We also prove that in ZFA, the "Multiple Choice" axiom implies the Krein-Milman axiom. We deduce that, in ZFA, the conjunction of the Hahn-Banach, Ingleton and Krein-Milman axioms does not imply the Axiom of Choice.
• Hyperplanes in matroids and the Axiom of Choice, Commentationes Mathematicae Universitatis Carolinae, (2022), vol. 63, issue 4, pp. 423-441.

Banach-Tarski decompositions

• Décomposition de Banach-Tarski d'un F2-ensemble libre en 4 morceaux, avec Hahn-Banach et une conséquence de l'axiome du choix dans les paires, Workshop LIM/LIRIS, 14-16 février 2023.
• Paradoxical decompositions of free F2-sets and the Hahn-Banach axiom, to appear in Mathematical Logic Quarterly.