On separators of the space of complete non-negatively curved metrics on the plane

TL;DR

This paper investigates the topological properties of the space of complete non-negatively curved metrics on the plane, showing it cannot be separated by certain infinite-dimensional sets and is continuum connected.

Contribution

It extends previous theorems by Belegradek and Hu, demonstrating new connectivity properties and separation limitations in the metric space and its moduli space.

Findings

The Hilbert cube cannot be separated by a weakly infinite dimensional subset.

The complement of such subsets in the metric space is continuum connected.

Results apply to the associated moduli space with certain Hausdorff set conditions.

Abstract

We shall prove that the Hilbert cube cannot be separated by a weakly infinite dimensional subset. As a corollary we obtain that the complement of a weakly infinite dimensional subset of the space of complete non negatively curved metrics is continuum connected. We can extend this result to the associated moduli space when the set removed is a Hausdorff space with Haver's property C. These results are refinements of theorems proven by Belegradek and Hu.