Transfinite Hausdorff dimension
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 the intermediate dimension property of {tHD}, we arrive for each countable ordinal number \(\alpha\) at a compact metric space \(X_\alpha\) such that tHD\((X_\alpha) = \alpha\) but which is zero-dimensional in the classical way.
scientific article; zbMATH DE number 5625475