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Results tagged with dimension-theory
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user 121665
Hausdorff dimension, box dimension, packing dimension and similar concepts.
12
votes
Existence of subset with given Hausdorff dimension
The answer is yes under the additional assumption that the set is compact and I do not know what happens in the general case. The result is a consequence of the following one, see [1] and references …
- 28k
12
votes
Is the complement of a zero-dimensional subset of the plane path-connected?
At least if $X$ is compact, the answer is yes. Indeed, by Corollary 2 of Theorem IV 3 in:
W. Hurewicz, H. Wallman, Dimension Theory. Princeton Mathematical Series, v. 4. Princeton University Press, …
- 28k
12
votes
Accepted
Hausdorff dimension of the graph of an increasing function
Theorem 1. The Hausdorff dimension of the graph $\Gamma_f$ of $f$ equals $1$.
Proof.
Take a partition of $[0,1]$ by intervals of length $1/n$. Since the function is increasing you can cover the g …
- 28k
4
votes
Doubling dimension vs other metric dimensions
Let ${\rm dim}_H X$ and ${\rm dim}_d X$ denote the Hausdorff and the doubling dimension respectively. It is easy so see that ${\rm dim}_H X\leq {\rm dim}_d X$. Indeed, if ${\rm dim}_d X=s$, then we …
- 28k