Generic Properties of Compact Metric Spaces

TL;DR

This paper demonstrates that within the Gromov-Hausdorff space of all compact metric spaces, a residual set exists where spaces exhibit both zero lower and infinite upper box dimensions, revealing surprising geometric properties.

Contribution

It establishes the existence of a residual subset in the Gromov-Hausdorff space with spaces having extreme box dimension properties, a novel geometric insight.

Findings

Residual subset exists with these properties

Spaces have zero lower box dimension

Spaces have infinite upper box dimension

Abstract

We prove that there is a residual subset of the Gromov-Hausdorff space (i.e. the space of all compact metric spaces up to isometry endowed with the Gromov-Hausdorff distance) whose points enjoy several unexpected properties. In particular, they have zero lower box dimension and infinite upper box dimension.