Making extensive use of small transfinite topological dimension trind, we
ascribe to every metric space $X$ an ordinal number (or $-1$ or $\Omega$)
$\text{tHD}(X)$, 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 $\text{tHD}(X_\alpha)=\alpha$ 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.