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Transfinite Hausdorff Dimension
- Speaker(s)
- Mariusz Urbański
- Affiliation
- University of North Texas
- Date
- Nov. 5, 2010, 10:15 a.m.
- Room
- room 5840
- Seminar
- Seminar of Dynamical Systems Group
Making extensive use of small transfinite topological dimension trind, we ascribe to every metric space $X$ an
ordinal number (or $-1$ or $\Om$) $\tHD(X)$, and we call it the transfinite Hausdorff dimension of $X$. This ordinal number shares many common features with Hausdorff dimension. It is monotone with respect to subspaces, it is invariant under bi-Lipschitz maps (but in general not under homeomorphisms), in fact like Hausdorff dimension, it does not increase under Lipschitz maps, and it also satisfies the intermediate dimension property. The primary goal of transfinite Hausdorff dimension is to classify metric spaces with infinite Hausdorff dimension.
As our main theorem, we show that for every countable ordinal number $\alpha$ there exists a compact metric space $X_\alpha$ (a subspace of the Hilbert space $l_2$) with $\tHD(X_\alpha)=\alpha$ and which is a topological Cantor set, thus of topological dimension $0$. In our proof we construct metric versions of Smirnov topological spaces and establish several properties of transfinite Hausdorff dimension, including its relations with classical Hausdorff dimension.
ordinal number (or $-1$ or $\Om$) $\tHD(X)$, and we call it the transfinite Hausdorff dimension of $X$. This ordinal number shares many common features with Hausdorff dimension. It is monotone with respect to subspaces, it is invariant under bi-Lipschitz maps (but in general not under homeomorphisms), in fact like Hausdorff dimension, it does not increase under Lipschitz maps, and it also satisfies the intermediate dimension property. The primary goal of transfinite Hausdorff dimension is to classify metric spaces with infinite Hausdorff dimension.
As our main theorem, we show that for every countable ordinal number $\alpha$ there exists a compact metric space $X_\alpha$ (a subspace of the Hilbert space $l_2$) with $\tHD(X_\alpha)=\alpha$ and which is a topological Cantor set, thus of topological dimension $0$. In our proof we construct metric versions of Smirnov topological spaces and establish several properties of transfinite Hausdorff dimension, including its relations with classical Hausdorff dimension.
