The $D$-Variant of Transfinite Hausdorff Dimension
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866929588905443328 |
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| author | Decker, Bryce Dalaklis, Nathan |
| author_facet | Decker, Bryce Dalaklis, Nathan |
| contents | 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 $0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_07454 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The $D$-Variant of Transfinite Hausdorff Dimension Decker, Bryce Dalaklis, Nathan General Topology 54F45 (Primary) 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 $0$. |
| title | The $D$-Variant of Transfinite Hausdorff Dimension |
| topic | General Topology 54F45 (Primary) |
| url | https://arxiv.org/abs/2411.07454 |