Topological classification of closed convex sets in Fréchet spaces

Taras Banakh, and Robert Cauty. "Topological classification of closed convex sets in Fréchet spaces." Studia Mathematica 205.1 (2011): 1-11. <http://eudml.org/doc/285892>.

@article{TarasBanakh2011,
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 $[0,1]ⁿ × [0,1)^\{m\} × ℓ₂(κ)$ for some cardinals 0 ≤ n ≤ ω, 0 ≤ m ≤ 1 and κ ≥ 0.},
author = {Taras Banakh, Robert Cauty},
journal = {Studia Mathematica},
keywords = {Fréchet and Hilbert space; convex set; separable; completely metrizable},
language = {eng},
number = {1},
pages = {1-11},
title = {Topological classification of closed convex sets in Fréchet spaces},
url = {http://eudml.org/doc/285892},
volume = {205},
year = {2011},
}

TY - JOUR
AU - Taras Banakh
AU - Robert Cauty
TI - Topological classification of closed convex sets in Fréchet spaces
JO - Studia Mathematica
PY - 2011
VL - 205
IS - 1
SP - 1
EP - 11
AB - 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 $[0,1]ⁿ × [0,1)^{m} × ℓ₂(κ)$ for some cardinals 0 ≤ n ≤ ω, 0 ≤ m ≤ 1 and κ ≥ 0.
LA - eng
KW - Fréchet and Hilbert space; convex set; separable; completely metrizable
UR - http://eudml.org/doc/285892
ER -