The $D$-Variant of Transfinite Hausdorff Dimension

Abstract

We assign every metric space $X$ the value $t_{D}HD(X)$, an ordinal number or one of the symbols $-1$ or $Ω$, 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 $t_{D}HD\geq ω_0$, then $HD(X) = \infty$. $t_{D}HD(X)<ω_1$ 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 $0$.