Topological classification of closed convex sets in Frechet spaces

TL;DR

This paper proves that non-separable convex subsets of Frechet spaces are homeomorphic to Hilbert spaces, solving a longstanding problem in infinite-dimensional topology and classifying all closed convex subsets in this context.

Contribution

It establishes that all non-separable convex subsets of Frechet spaces are homeomorphic to Hilbert spaces, resolving a 30-year-old open problem.

Findings

Non-separable convex subsets of Frechet spaces are homeomorphic to Hilbert spaces.

All closed convex subsets of Frechet spaces are classified as products involving [0,1]^n, [0,1)^m, and l_2(k).

The result completes the topological classification of convex sets in infinite-dimensional spaces.

Abstract

We prove that each non-separable completely metrizable convex subset of a Frechet space is homeomorphic to a Hilbert space. This resolves an old (more than 30 years) problem of infinite-dimensional topology. Combined with the topological classification of separable convex sets due to Klee, Dobrowoslki and Torunczyk, this result implies that each closed convex subset of a Frechet space is homemorphic to $[0,1]^n\times [0,1)^m\times l_2(k)$ for some cardinals $0\le n\le\omega$, $0\le m\le 1$ and $k\ge 0$.