The question: Can a closed simple 2D curve (by 2D curve, I mean a 1-dimensional continuum on a 2-dimensional plane) be a less-than-1-dimensional fractal (by dimension, I mean Hausdorff dimension) ?
I've searched through some famous fractals, none of them satisfy. Maybe there is a theorem that prevents the answers, but I'm too new to topology to notice.
Thanks in advance.