  We introduce a new concept of dimension for metric spaces, the so-called
topological Hausdorff dimension. It is defined by a very natural combination of
the definitions of the topological dimension and the Hausdorff dimension. The
value of the topological Hausdorff dimension is always between the topological
dimension and the Hausdorff dimension, in particular, this new dimension is a
non-trivial lower estimate for the Hausdorff dimension.
  We examine the basic properties of this new notion of dimension, compare it
to other well-known notions, determine its value for some classical fractals
such as the Sierpinski carpet, the von Koch snowflake curve, Kakeya sets, the
trail of the Brownian motion, etc.
  As our first application, we generalize the celebrated result of Chayes,
Chayes and Durrett about the phase transition of the connectedness of the limit
set of Mandelbrot's fractal percolation process. They proved that certain
curves show up in the limit set when passing a critical probability, and we
prove that actually `thick' families of curves show up, where roughly speaking
the word thick means that the curves can be parametrized in a natural way by a
set of large Hausdorff dimension. The proof of this is basically a lower
estimate of the topological Hausdorff dimension of the limit set. For the sake
of completeness, we also give an upper estimate and conclude that in the
non-trivial cases the topological Hausdorff dimension is almost surely strictly
below the Hausdorff dimension.
  Finally, as our second application, we show that the topological Hausdorff
dimension is precisely the right notion to describe the Hausdorff dimension of
the level sets of the generic continuous function (in the sense of Baire
category) defined on a compact metric space.
