Topological classification of closed convex sets in Fréchet spaces
Volume 205 / 2011
Studia Mathematica 205 (2011), 1-11
MSC: Primary 57N17; Secondary 46A55.
DOI: 10.4064/sm205-1-1
Abstract
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 $\kappa \ge 0$.
