Questions tagged [dimension-theory]

Hausdorff dimension, box dimension, packing dimension and similar concepts.

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3 answers
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Area of the boundary of the Mandelbrot set ?

My second question about Shishikura's result : Shishikura (1991) proved that the Hausdorff Dimension of the boundary of the Mandelbrot set equals 2, in this paper 1. In a sense, could we consider it ...
Alexis Monnerot-Dumaine's user avatar
52 votes
3 answers
5k views

What the heck is the Continuum Hypothesis doing in Weibel's Homological Algebra?

On page 98 of Weibel's An Introduction to Homological Algebra he mentions that the ring $R = \prod_{i=1}^\infty \mathbb{C}$ has global dimension $\geq 2$ with equality iff the continuum hypothesis ...
David White's user avatar
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30 votes
1 answer
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Rank of a module

What's wrong with defining the rank of a finitely generated module over any (commutative) ring to be just the smallest number of generators? All books I know define rank only locally this way. But why ...
ashpool's user avatar
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5 votes
0 answers
225 views

Do $G_\delta$-measurable maps preserve dimension?

This question (in a bit different form) I leaned from Olena Karlova. Question. Let $f:X\to Y$ be a bijective continuous map between metrizable separable spaces such that for every open set $U\subset ...
Taras Banakh's user avatar
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26 votes
2 answers
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Analogues of Luzin's theorem

If $X$ is a compact metric space and $\mu$ is a Borel probability measure on $X$, then the space $C(X)$ of continuous real-valued functions on $X$ is a closed nowhere dense subset of $L^\infty(X,\mu)$,...
Vaughn Climenhaga's user avatar
9 votes
2 answers
566 views

Unknown work of Nöbeling on topological/Hausdorff dimension

Let $\mathcal{H}^n$ denote the Hausdorff measure, $\dim_H X$ the Hausdorff dimension, and $\dim X$ the topological dimension of $X$. A well known result of Szpilrajn (He changed his name to ...
Piotr Hajlasz's user avatar
7 votes
1 answer
1k views

Hausdorff dimension of the graph of an increasing function

Let $f$ be a continuous, strictly increasing function from $[0,1]$ to itself with $f(0)=0, f(1)=1$. Let $\Gamma_f$ denote its graph. What can be said about the Hausdorff dimension of $\Gamma_f$? In ...
Nikita Sidorov's user avatar
7 votes
3 answers
651 views

How can dimension depend on the point?

Let $M$ be a metric space. For any subset $A\subset M$ let $\dim(A)$ denote its Hausdorff dimension. For $x\in M$, define the dimension of $M$ at $x$ by $\dim(x)=\lim_{r\to0}\dim(B(x,r))$; this limit ...
Joonas Ilmavirta's user avatar
23 votes
3 answers
1k views

Existence of subset with given Hausdorff dimension

Let $A\subseteq \mathbb{R}$ be Lebesgue-measurable and let $0<\alpha<1$ be its Hausdorff dimension. For a given $0<\beta <\alpha$ can we find a subset $B\subset A$ with Hausdorff ...
Severin Schraven's user avatar
14 votes
6 answers
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More upper/lower semi-continuous functions in (algebraic) geometry?

The notion of upper/lower semi-continuity is sometimes encountered in algebraic geometry. Here by upper semi-continuity one means a function on a topological space $f:X\rightarrow S$ with value in ...
12 votes
3 answers
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Can we say anything about the Krull dimension of a localization?

I'm looking for a theorem of the form If $R$ is a nice ring and $v$ is a reasonable element in $R$ then Kr.Dim$(R[\frac{1}{v}])$ must be either Kr.Dim$(R)$ or Kr.Dim$(R)-1$. My attempts to do ...
David White's user avatar
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11 votes
1 answer
525 views

Can dividing out a group action can increase the Lebesgue dimension ?

Given any space $X$ of Lebesgue dimension at most $n$. Suppose a group $G$ acts on $X$ continuously. Can the dimension of the quotient $G\backslash X$ exceed the dimension of $X$? I know examples, ...
HenrikRüping's user avatar
9 votes
3 answers
800 views

When is "metric dimension" well defined?

A subset $B$ of a metric space $(M,d)$ is called a metric generating set if and only if $$[\forall b \in B, d(x,b)=d(y,b)] \implies x = y \,. $$ A metric generating set $B$ is called a metric basis ...
Chill2Macht's user avatar
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8 votes
3 answers
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How big can the Hausdorff dimension of a function graph get?

This question is inspired by How kinky can a Jordan curve get? What is the least upper bound for the Hausdorff dimension of the graph of a real-valued, continuous function on an interval? Is the ...
Harald Hanche-Olsen's user avatar
8 votes
1 answer
226 views

Can a smooth diffeomorphism of a Riemannian manifold have only positive Lyapunov exponents on a large set?

Let $M$ be a compact Riemannian manifold, $f: M \to M$ a diffeomorphism, and $\mu$ an ergodic measure for $M$. Suppose that the support of $\mu$ is not a finite set. Is it possible that all the ...
Will Sawin's user avatar
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7 votes
2 answers
2k views

Haar measure on the Grassmannian space

The grassmannian space $G(n,m)$ may be identified with the quotient space $O(n)/(O(m)\times O(n-m)$. As such, it is endowed with a natural invariant probability measure which I call "Haar measure on $...
timofei's user avatar
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6 votes
1 answer
414 views

Transitive homeomorphisms of Erdős spaces

A surjective homeomorphism $h:X\to X$ is minimal if $$\overline{\{h^n(x):n\in \mathbb N\}}=X$$ for every $x\in X$. In other words, the orbit of each point is dense. Does either of the Erdös spaces $\...
D.S. Lipham's user avatar
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6 votes
1 answer
476 views

Is there an almost strongly zero-dimensional space which is not strongly zero-dimensional

A Tychonoff space $X$ is called strongly zero-dimensional if each functionally closed subset $F$ of $X$ is a $C$-set, which means that $F$ is the intersection of a sequences of clopen sets in $X$. A ...
Lviv Scottish Book's user avatar
6 votes
1 answer
328 views

Factorization of a certain map through a CW-complex

Suppose that $X$ is a paracompact Hausdorff space (e.g. a metric space) with $\dim X=n$ (the Lebesgue covering dimension). I want to find a proof (or a reference) that any (continuous) map $f: X \to K(...
Ilja's user avatar
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6 votes
0 answers
124 views

Is there a hereditarily disconnected space which is not the union of countably many totally disconnected subspaces?

A topological space $X$ is called $\bullet$ totally disconnected if for any distinct points $x,y\in X$ there exists a clopen set $U\subseteq X$ such that $x\in U$ and $y\notin U$; $\bullet$ ...
Taras Banakh's user avatar
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5 votes
1 answer
277 views

Is the Hilbert cube the countable union of punctiform spaces?

Recall that a (separable) metric space is called punctiform, if all its compact subspaces are zero-dimensional. While "natural" spaces would seem to be punctiform if they already themselves ...
Arno's user avatar
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3 votes
0 answers
321 views

What is the connection between the Riemann Xi-function and n-sphere? [closed]

Riemann's Xi-function is defined as $$\xi(s) = \pi^{-s/2}\ \Gamma\left(\frac{s}{2}\right)\ \zeta(s)$$ At the same time we have the following formulas for n-sphere's area and volume: $$\begin{array}{...
Anixx's user avatar
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3 votes
1 answer
148 views

Topological spaces with Lebesgue covering dimension 1

We know that all connected subsets of $\mathbb{R}$( with the usual topology) has no empty interior. I would like to know if this fact remains true for a general connected topological space with the ...
Didi's user avatar
  • 95
2 votes
1 answer
387 views

Extremally disconnectedness and 0-dimensional space

Let $X$ be a non-empty topological space. Then we have the following concepts for the topological space $X $: 1) We say $X $ has property $*$, if for every closed subset $A$ of $X$ and every open ...
Alesix's user avatar
  • 21
1 vote
1 answer
224 views

Quotients of the irrationals

Everyone knows that there is a closed equivalence relation $\sim$ on the Cantor set $C$ such that each non-trivial equivalence class has exactly $2$ points and $[0,1]\simeq C/\sim$. Thus a closed ...
D.S. Lipham's user avatar
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