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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
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
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