The $D$-Variant of Transfinite Hausdorff Dimension

TL;DR

This paper introduces a new ordinal-valued dimension function for metric spaces, called the $D$-variant of transfinite Hausdorff dimension, which improves classification of spaces with infinite Hausdorff dimension.

Contribution

It defines the $D$-variant of transfinite Hausdorff dimension, addressing limitations of previous transfinite dimensions, and explores its properties and implications for classifying metric spaces.

Findings

$t_{D}HD$ is monotone and bi-Lipschitz invariant.

$t_{D}HD eq ext{finite}$ implies infinite Hausdorff dimension.

For separable spaces, $t_{D}HD$ is always less than $ ext{omega}_1$.

Abstract

We assign every metric space $X$ the value $t_{D}HD(X)$, an ordinal number or one of the symbols $-1$ or $\Omega$, and we call it the $D$-variant of transfinite Hausdorff dimension of $X$. This ordinal assignment is primarily constructed by way of the $D$-dimension, a transfinite dimension function consistent with the large inductive dimension on finite dimensional metric spaces while also addressing shortcomings of the large transfinite inductive dimension. Similar to Hausdorff dimension, $t_{D}HD(\cdot)$ is monotone with respect to subspaces, and is a bi-Lipschitz invariant. It is also non-increasing with respect to Lipschitz maps and satisfies a coarse intermediate dimension property. We also show that this new transfinite Hausdorff dimension function addresses the primary goal of transfinite Hausdorff dimension functions; to classify metric spaces with infinite Hausdorff dimension. In particular, we show that if $t_{D}HD\geq \omega_0$, then $HD(X) = \infty$. $t_{D}HD(X)<\omega_1$ for any separable metric space, and that one can find a metrizable space with $t_{D}HD(X)$ bounded between a given ordinal and it's successive cardinal with topological dimension $0$.