Compact covering mappings and cofinal families of compact subsets of a Borel set

G. Debs¹ ; J. Saint Raymond²

¹ Equipe d'Analyse Université Paris 6 Boîte 186 4, place Jussieu 75252 Paris CEDEX 05, France
² Equipe d'Analyse Université Paris 6 Boite 186 4, place Jussieu 75252 Paris CEDEX 05, France

Fundamenta Mathematicae, Tome 167 (2001) no. 3, pp. 213-249

Résumé

Among other results we prove that the topological statement “Any compact covering mapping between two ${\bf \Pi }^0_{3}$ spaces is inductively perfect” is equivalent to the set-theoretical statement “$\forall \alpha \in \omega ^\omega ,$ $\omega _1^{L(\alpha )}\omega _1$”; and that the statement “Any compact covering mapping between two coanalytic spaces is inductively perfect” is equivalent to “Analytic Determinacy”. We also prove that these statements are connected to some regularity properties of coanalytic cofinal sets in ${\cal K}(X)$, the hyperspace of all compact subsets of a Borel set $X$.

Détail

DOI : 10.4064/fm167-3-2

Keywords: among other results prove topological statement compact covering mapping between spaces inductively perfect equivalent set theoretical statement forall alpha omega omega omega alpha omega statement compact covering mapping between coanalytic spaces inductively perfect equivalent analytic determinacy prove these statements connected regularity properties coanalytic cofinal sets cal hyperspace compact subsets borel set