Transfinite Hausdorff dimension
type: Item
instance of: scholarly article
title: Transfinite Hausdorff dimension
zbMATH Open document ID: 1181.54040
author: Mariusz Urbański
published in: Topology and its Applications
publication date: 2009-11-05T00:00:00Z
review text: The author develops the concept of transfinite Hausdorff dimension. The primary goal of this dimension is to classify metric spaces with infinite Hausdorff dimension. Felix Hausdorff defined the concept of Hausdorff dimension, \(HD(X,\rho)\) of a metric space \((X,\rho)\). This dimension is not a topological invariant, but an invariant under a Lipschitz mapping. In the finite case the small inductive dimension ind(X) and the Hausdorff dimension are related as ind\((X) = \inf\{HD(X,\rho)\}\) where the infimum is taken over all admissible metrics \(\rho\) on the space \(X\). Using the small transfinite inductive dimension, trind, Urbański defines the transfinite Hausdorff dimension, tHD\((X)\) by letting tHD\((X) \leq \alpha\) if and only if trind \((Y) \leq \alpha\) for every space \(Y\) which is the image of \(X\) under a Lipschitz map \(f : X\to Y\). In the finite case we have no equivalence of values. But in the finite case we have that trind\((X) \leq\) tHD\((X) \leq\) integer value\((HD(X))\). Therefore, whenever tHD\((X)\geq \omega_o\), we have \(HD(X)=+\infty\). For every separable metric space \(X\) with existing tHD\((X)\) we have tHD\((X)< \omega_1\). Various basic properties of the new dimension function are proven, like the monocity under metric subspaces inheriting the metric. Also the intermediate dimension property and the lower sum theorem and other sum theorems are proven. An example is the following result: When a compact metric space X is the union of a closed set \(X_0\) and a locally finite closed collection \(\{X_{j}\}_{j \in J}\) in \(X - X_0\), then tHD\((X) \leq (1+\text{tHD}(X_0)\) * (lsup\(\{\text{tHD}(X_{j}):j \in J\})\). An important result is the result on the Smirnov like Cantor set \(C_\alpha\) contained in the Smirnov Space \(S_\alpha\). This space is proven to have tHD\((C_\alpha) \geq\) trind\((S_\alpha)\) and also ind\((C_\alpha)=0\), stressing the difference between the dimension functions {trind} and {tHD}. Using this result and …
reviewed by: Piet Borst
Mathematics Subject Classification ID: 54F45
Mathematics Subject Classification ID: 30D05
zbMATH DE Number: 5625475
zbMATH Keywords: transfinite Hausdorff dimension
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: Lipschitz continuous functions
zbMATH Keywords: small transfinite topological dimension
zbMATH Keywords: metric spaces
zbMATH Keywords: ordinal numbers
MaRDI profile type: Publication
full work available at URL: j.topol.2009.01.025
OpenAlex ID: W2059656612
cites work: Q4441888
cites work: Q4855460
cites work: The Realization of small Distances in Plane Sets of Positive Measure
cites work: Q3714302
cites work: Q4328334
cites work: On dimensional properties of infinite dimensional spaces
DOI: 10.1016/J.TOPOL.2009.01.025
description: scientific article; zbMATH DE number 5625475