# Tag Info

## Hot answers tagged dimension-theory

22 votes
Accepted

### Dimension in CW-approximation

Barratt and Milnor (An Example of Anomalous Singular Homology) proved that (for $n > 1$) the singular homology of the union of countably many $n$-spheres with one point in common and radii tending ...
• 7,596
21 votes
Accepted

### Why do almost all points in the unit interval have Kolmogorov complexity 1?

I'm not an expert on Kolmogorov complexity, but this does seem like a counting argument: for any fixed $\epsilon > 0$, there are only $\sum_{i = 1}^{(1-\epsilon)n} 2^i < 2^{(1-\epsilon)n+1}$ ...
• 2,313
20 votes
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### Two definitions of Lebesgue covering dimension

As you refer to Engelking's "Dimension theory" book, I suppose you know the following two statements, but anyway, here they are: The notions agree for separable metric spaces, by Exercise 1.7.E of ...
• 17.2k
20 votes
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### A set whose Hausdorff dimension gradually changes?

I assume you want a set $A\subseteq [0,1]$ such that $\dim (A\cap [0,x])=x$ for all $x$. We can define $A_1$ by taking the union of a (Borel) subset of dimension $0$ of $[0,1/2]$ with a subset of ...
• 22.3k
14 votes
Accepted

### Existence of subset with given Hausdorff dimension

First of all, $\dim_{H} (A) = \alpha$ iff $H^k(A)=\infty$ for all $k<\beta$ and $H^k(A) = 0$ for all $k>\beta$. Then $H^\alpha(A) = \infty$ for all $\alpha \in (0,\beta)$. If $A$ is closed ...
• 1,277
14 votes
Accepted

### Is the complement of a zero-dimensional subset of the plane path-connected?

If the zero-dimensional set $X$ is not closed, then the answer is "no". To construct a suitable example, take any open bounded neighborhood $U\subset\mathbb R^2$ of zero, whose boundary $\partial U$ ...
• 40.4k
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 graph ...
• 26.9k
12 votes
Accepted

### If $\text{dim}(X \times X) = 2\text{dim}(X)$, does $\text{dim}(X^n) = n\text{dim}(X)$?

As John Samples noted in his comment, Dranishnikov's Theory of cohomological dimension implies the positive answer to this problem for compact (even $\sigma$-compact) metrizable spaces. Namely, ...
• 40.4k
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 ...
• 26.9k
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, ...
• 26.9k
12 votes
Accepted

### Unknown work of Nöbeling on topological/Hausdorff dimension

So, the sought for paper is: Nöbeling, G., Hausdorffsche und mengentheoretische Dimension, Ergebnisse math. Kolloquium Wien 3, 24-25 (1931). And here is a translation" (to English and to modern ...
10 votes

• 26.9k
7 votes

### What is special to dimension 8?

Some special properties of dimension 8, in addition to the ones you identify: Bernstein's problem holds up to dimension $n=8$. The only function of $\mathbb{R}^{n-1}$ whose graph in $\mathbb{R}^n$ ...
7 votes

### What is special to dimension 8?

The following is too long to leave as a comment, so I post it as an answer. At least for the cases of the Bernstein problem for minimal graphs, Simons' theorem on stable minimal cones, etc., there is ...
7 votes

### Fraction dimensional "Euclidean" space

One remark: there is no metric space $\mathbb{E}^{1/2}$ of Hausdorff dimension $\frac{1}{2}$ whose topological square $\mathbb{E}^{1/2} \times \mathbb{E}^{1/2}$ is homeomorphic to $\mathbb{E}^1$, i.e.,...
• 344
7 votes
Accepted

### Does finite Hausdorff dimension imply finite packing dimension?

A construction used (repeatedy) in the paper Edgar, G. A., Centered densities and fractal measures, New York J. Math. 13, 33-87 (2007). ZBL1112.28004. For more information, see that paper. We ...
• 40.1k
6 votes
Accepted

### Is there a gap between the Hausdorff and the lower Minkowski dimensions?

I believe what you are asking is whether the Hausdorff dimension can be strictly less than the lower packing dimension (also called the lower modified box dimension). I'm pretty sure the strongest ...
• 3,107
6 votes

### Fractals of dimension zero

Closed uncountable sets of Hausdorff dimension zero turn up from time to time in practice (one of my papers with Nikita Sidorov features one) but I think that there does not exist a set which is ...
• 6,166
6 votes
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### Can a smooth diffeomorphism of a Riemannian manifold have only positive Lyapunov exponents on a large set?

The answer is no. In fact, the following result holds: if $M$ is a compact manifold, $f\in\mathrm{Diff}^{1+\alpha}(M)$ and $\mu$ is an ergodic $f$-invariant probability measure such that all its ...
• 865
6 votes
Accepted

### Doubling dimension vs other metric dimensions

In one direction, a rapidly branching tree will have very high doubling dimension, while having topological dimension $0$ (or $1$, if you include the edges). In another direction there is a bound, and ...
• 95.5k
6 votes
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### Real rank 0 implies stable rank 1 on $C^\ast$-algebras?

There are "cheap" examples. If a C*-algebra $A$ has stable rank 1 then it is stably finite. On the other hand every simple, unital purely infinite C*-algebra has real rank 0. So every simple, ...
• 2,669

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