Making extensive use of small transfinite topological dimension trind, we
ascribe to every metric space X an ordinal number (or -1
or Ω) 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 α
there exists a compact metric space Xα (a subspace
of the Hilbert space l2)
with tHD(Xα)=α 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.
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