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 for some cardinals 0 ≤ n ≤ ω, 0 ≤ m ≤ 1 and κ ≥ 0.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm205-1-1,
author = {Taras Banakh and Robert Cauty},
title = {Topological classification of closed convex sets in Fr\'echet spaces},
journal = {Studia Mathematica},
volume = {204},
year = {2011},
pages = {1-11},
zbl = {1234.57027},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm205-1-1}
}
Taras Banakh; Robert Cauty. Topological classification of closed convex sets in Fréchet spaces. Studia Mathematica, Tome 204 (2011) pp. 1-11. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm205-1-1/