Topological classification of closed convex sets in Fréchet spaces

Taras Banakh ¹   ; Robert Cauty ²

¹ Wydział Matematyczno-Przyrodniczy Uniwersytet Jana Kochanowskiego Świętokrzyska 15 25-406 Kielce, Poland and Department of Mathematics Ivan Franko National University of Lviv Universytetska 1 79000 Lviv, Ukraine
² Faculté de Mathématiques Université Paris VI 4, place Jussieu 75005 Paris, France

Studia Mathematica, Tome 205 (2011) no. 1, pp. 1-11

Résumé

We prove that each non-separable completely metrizable convex subset of a Fréchet space is homeomorphic to a Hilbert space. This resolves a more than 30 years old problem of infinite-dimensional topology. Combined with the topological classification of separable convex sets due to Klee, Dobrowolski and Toruńczyk, this result implies that each closed convex subset of a Fréchet space is homeomorphic to $[0,1]^n\times [0,1)^m\times \ell _2(\kappa )$ for some cardinals $0\le n\le \omega $, $0\le m\le 1$ and $\kappa \ge 0$.

Détail

DOI : 10.4064/sm205-1-1

Mots-clés : prove each non separable completely metrizable convex subset chet space homeomorphic hilbert space resolves years old problem infinite dimensional topology combined topological classification separable convex sets due klee dobrowolski toru czyk result implies each closed convex subset chet space homeomorphic times times ell kappa cardinals omega kappa