1

I have a rather basic question but I couldn't find a straight answer anywhere. Consider $C \subset [0,1]$ a Cantor set, that is a closed, nowhere dense and perfect subset of $[0,1]$.

Is the Hausdorff dimension of $C$ necessarily positive?

And even more basic: is it possible to find $A$ an uncountable subset of $[0,1]$ whose Hausdorff dimension is $0$? Obviously a negative answer to the former question would provide such an example.

I'd be very satisfied with dry answers with references if that's all you can provide :)

Selim Ghazouani
  • 2,312
  • 16
  • 26

1 Answers1

1

You may construct a Cantor set of 0 Hausdorff dimension. In the standard construction you take away the middle 1/3 at every step. If instead you take away the middle $n/(n+1)$ at the $n$'th step the dimension will be zero. Just calculate the box dimension.

H. H. Rugh
  • 35,992