Von Neumann in his book Continuous Geometry introduced (in a suitable lattice) a dimension function that has a continuous range. The definition of a dimension function is axiomatic: see Continuous geometry on Wikipedia. It can take non-integer values; for example, the entire interval $[0,1]$ can be the range of a dimension function.

Also the Hausdorff dimension has a continuous range and is defined on the lattice of the subsets of a metric space. Is there any relation between these two concepts of dimension?

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    $\begingroup$ I can say "there is no known relation". But to say "there is no relation" is beyond my pay grade. $\endgroup$ – GEdgar Nov 18 '14 at 14:41
  • $\begingroup$ @GEdgar I found this, but it's too hard for me. $\endgroup$ – Emilio Novati Dec 5 '14 at 16:32

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