I can't comment on the "Hausdorff Dimension", but with regards to the 11 dimensional spacetime of string and "M"Theories:
The concept of folded/compactified (what's wrong with "compact"?) dimensions is based on a boundaryless, finite spacetime. Typically, something like the 2D surface of a 3D torus, but enviosioned in a higher dimensional spacetime.
In considering such geometries, though, particularly within the advent of string theories, the reasoning behind the refinement to exactly 3+1+7 dimensions is born from the need to break symmetry of the E8xE8 group to resolve gluons and electroweak force. Only the E8xE8 group of heterotic strings encompassed gravity in a stable, ten-dimensional spacetime.
To reconcile the neceessary asymetry for the strong and weak forces, required six of these dimensions to be compacted. The reasons for this are closely defined by observed chirality, mass, and topological considerations.
The process is painstaking, and still has many formulae that cannot yet be fully solved, although in a methodical approach, starting with the lowest dimensionality where all elementary objects are bosoinic and chiral symmetry is ratified, there are 4n+2 dimensions. So at n=0, we have 2 extra dimensions. Clearly, this is too few, but the next, when n=1 yields six.
To understand more of thew reasoning behind this, the topology of the dimensions is defined by its 'Euler Characteristic'. The Euler characteristic of the orbifold Calabi-Yau) space determines the generations of fermionic objects, such as neutrinos. Since there are, to the best of our knowledge, three such generations, only certain Euler-characterised Calabi-Yau spaces are valid for describing the observed universe.
String theory in particular contains a much-needed property in defining this description of spacetime wherewby closed-loops can become 'trapped' only by topologies with "holes" (or singularities) and allowing for effects of the curled-up (compact) dimensions manifest effects in our expanded observable spacetime.
Roger Penrose' Twistor space is also based on a ten-dimensional spacetime, and bears many similarities in terms of interaction with string theory, although not based on extended one dimensional objects. Penrose makes use of complex dimensionality in what he terms 'twistor space', where Twistors are complex dimensional objects that can form vast networks which themselves define spacetime geometry and bridge the gap between local and non-local effects.
The 11th Dimension is added only where the development of unified string theories gives rise to brane theories or M-Theory, where the eleventh dimension is a higher, expanded spatial dimension in which separate "brane" universes reside.
