If we define a curve as a continuos image of the (unit) interval then one can construct example of so called Peano curve which is the whole square. However if our curve is homeomorphic to the (unit) interval (let us call such a curve by arc) then such situation cannot occur-the dimension is the topological invariant. However recently I've heard that still one can construct arcs of positive measure. How those "big" arcs are constructed?
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https://en.wikipedia.org/wiki/Osgood_curve – Dec 07 '15 at 21:49