Questions tagged [dimension-theory]
Hausdorff dimension, box dimension, packing dimension and similar concepts.
161
questions
2
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0answers
71 views
Dimension of Cartesian products
Is there a notion of dimension such that for all Borel sets $A,B\subseteq\mathbb{R}^{n}$ we have
$$ \dim(A\times B)=\dim(A)+\dim(B)?$$ For topological, Minkowski, packing and Hausdorff dimension this ...
2
votes
1answer
125 views
Embedding CW-complexes into infinite-dimensional topological vector spaces
Sometimes it is desirable to embed CW-complexes into real vector spaces, to use a simple linear algebra to work with them. Result on embedding into Euclidean spaces are well known, check Hatcher‘s ...
1
vote
0answers
26 views
Selection theorems and homeomorphism groups
All spaces are separable metrizable.
Let $f:X\to Y$ be a continuous open mapping from a Polish space $X$ onto a zero-dimensional space $Y$. By the Michael selection theorem (zero-dimensional version), ...
1
vote
0answers
80 views
Hausdorff dimension and critical exponent of words
What is the Hausdorff dimension of the subset $S_c \subset [0,1]$ of points such that the critical exponent of their binary expansion is $c$? It's clear that $\dim_H S_{\infty}=1$, but what can be ...
4
votes
0answers
233 views
Investigating a 1/2 -dimensional sphere and defining a fractional Euclidean space
A small note
I'm a new member on MO and I'm not sure if this question fits in here. If not, please don't be too hard on me. I am transferring a question I asked on MSE in here, because the user who ...
2
votes
0answers
60 views
Increasing a nowhere dense set in $\mathfrak E_{\mathrm{c}}$
Let $X$ be a closed nowhere dense subset of the complete Erdos space $$\mathfrak E_{\mathrm{c}}=\{x\in \ell^2:x_n\notin \mathbb Q\text{ for all }n<\omega\}.$$
Can you always find a closed nowhere ...
1
vote
0answers
72 views
Equidimensional Morphism
I am reading the paper "Relative Cycles and Chow Sheaves" due to Suslin and Voevodsky. Here we have the following definition:
Definition 2.1.2.
A morphism of schemes $p:X\rightarrow S$ is ...
3
votes
0answers
70 views
Isotopy Classes and Embeddability of Products in $\mathbb{R}^2$
On MSE I asked if the plane contains an uncountable collection of mutually disjoint copies of the Warsaw Circle; it seems to be false, and is probably already known but I'm not sure that anybody has ...
2
votes
1answer
75 views
Control on dimension of image
Let $f:E\rightarrow F$ be a map between Banach spaces E and F; E finite dimensional (>0) and F infinite dimensional. Let $F$ be equipped with its weak topology and suppose that $f$ is strong-weak ...
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votes
0answers
37 views
Closed set with full box dimension and non-full Hausdorff dimension
Tried posting this on math.SE first but didn't get any responses.
Is there an example of a closed set $E \subseteq [0,1]$ such that $\dim_B(E) = 1$ and $\dim_H(E) < 1$? Here $\dim_B$ is box (...
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0answers
22 views
On the dimension of the projected rank $r$ matrix space
In $d$-dimensional complex number space $\mathbb{C}^{ d}$, we can define the rank at most $r$ matrix space
$$
S=\{A|\ \mathrm{rank}(A)\leq r\}\subseteq \mathbb{C}^{d\times d}.
$$
The dimension of ...
6
votes
0answers
79 views
Existence of stable spaces
An element $X$ of a class of topological spaces is called the stable space for that class if for every space $Y$ in the class we have that $X\times Y$ is homeomorphic to $X$. Note that a stable space ...
6
votes
1answer
118 views
Subset of $\mathbb R$ with equal Fourier, Hausdorff and Minkowski dimensions
It is a standard fact that for $0\leq s\le1$, there is a compact set $C\subseteq [0,1]$ with Hausdorff and Minkowski dimensions $s$ (by modifying the construction of a Cantor set).
It is also a ...
0
votes
1answer
83 views
Can someone please help me understand the concept of twins? [closed]
I am unable to understand Lemma 2.3 of
Carmen Hernando, Mercè Mora, Ignacio M. Pelayo, Carlos Seara, David R. Wood, Extremal Graph Theory for Metric Dimension and Diameter, Electronic J. ...
5
votes
0answers
138 views
Which metrizable spaces can be embedded into the countable power of $\omega$ with the cofinite topology?
Let $\omega_{cf}$ be the countable space $\omega=\{0,1,2,3,\dots\}$ endowed with the cofinite topology $$\tau_{cf}=\{\emptyset\}\cup\{U\subseteq\omega:\omega\setminus U\mbox{ is finite}\}.$$
It is ...
1
vote
0answers
95 views
Minkowski (box-counting) dimension of generalized Cantor set
I'm trying to solve this problem.
For $0<\alpha, \beta<1,$ let $K_{\alpha, \beta}$ be the Cantor set obtained as an intersection of the following nested compact sets. $K_{\alpha, \beta}^{0}=[0,...
5
votes
1answer
219 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 ...
5
votes
2answers
238 views
How to understand the “boundary” of subscheme, as defined in “An elementary characterisation of Krull dimension”
In An elementary characterisation of Krull dimension and A short proof for the Krull dimension of a polynomial ring, Coquand, Lombardi, and Roy give an elementary characterization of Krull dimension, ...
7
votes
2answers
449 views
A natural $\mathbb Q\times \mathbb P$ subset of $\mathbb R$?
I would like a simple description of a dense subset of $\mathbb R$ which is homeomorphic to $\mathbb Q\times \mathbb P$. Preferably the description will be of an algebraic nature, and perhaps the set ...
0
votes
0answers
28 views
What is the analog of the symmetrized Jacobi matrix for delay equations?
For a linear system of ODEs in $\mathbb{R}^{n}$ (with the usual inner product), say $\dot{V}(t) = A(t) V(t)$, we know that if $\xi_{1},\ldots,\xi_{k} \in \mathbb{R}^{n}$ and $V_{j}(t)=V_{j}(t,\xi_{j})$...
4
votes
1answer
156 views
$\varepsilon$-net of a $d$-dimensional unit ball formed by power set of $V = \{+1, 0 -1\}^d$
I have a set of $d$-dimensional vectors $V = \{+1, 0, -1\}^d $. Then $P(V)$ constitutes the power set of $V$. I now construct a set of unit vectors $V_{\mathrm{sum}}$ from the power set $P(V)$ such ...
8
votes
2answers
447 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 ...
24
votes
1answer
1k views
What is special to dimension 8?
Dimension $8$ seems special, as the partial list below might indicate.
Is there any overarching reason that dim-$8$ is "more special" than, say, dim-$9$?
Surely it isn't it, in the end, simply because ...
1
vote
1answer
83 views
Perfect images of complete Erdős space
Let $\mathbb P$ denote the space of irrational numbers. In an answer to this question, Taras Banakh showed that the perfect images of $\mathbb P$ are precisely the Polish spaces with no compact ...
2
votes
1answer
183 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 ...
2
votes
0answers
72 views
Fourier dimension of radial set
In his 1967 article "Sur un theoreme de R. Salem", Gatesoupe proved that if a set $A\subset [0,1]$ has Fourier dimension $\alpha$ then the set $\tilde A:=\{x\in \mathbb{R}^n: |x| \in A\}$ has Fourier ...
4
votes
0answers
414 views
The “core” of complete Erdős space
This question is about the Erdős spaces:
$\mathfrak E=\{x\in \ell^2:x_n\in \mathbb Q\text{ for all $n<\omega$}\}$; and
$\mathfrak E_c=\{x\in \ell^2:x_n\in \mathbb P\text{ for all $n<\omega$}\}...
10
votes
2answers
782 views
Is the complement of a zero-dimensional subset of the plane path-connected?
Let $X$ be a zero-dimensional subset of the plane $\mathbb R ^2$. Is $\mathbb R ^2\setminus X$ necessarily path-connected? I feel the answer must be yes but I need a reference. If it helps, assume $...
5
votes
1answer
141 views
A classification of $G_{\delta\sigma}$ zero-dimensional spaces?
Among separable metrizable spaces:
Cantor set is the unique compact zero-dimensional space without isolated points.
$\mathbb Q$ is the unique countable space without isolated points
$\mathbb R \...
9
votes
0answers
95 views
Real Rank of $M_n(A)$
The real rank for C*-algebras was defined by Brown-Pedersen in [1] as a noncommutative analog of covering dimension. Given a unital C*-algebra $A$, its real rank $\mathrm{rr}(A)$ is the smallest ...
3
votes
2answers
886 views
A set whose Hausdorff dimension gradually changes?
Can there be a set whose Hausdorff dimension gradually changes?
For instance, a set of real numbers contained in an interval, whose Hausdorff dimension is 0 at the beginning and 1 closer to the end, ...
10
votes
1answer
719 views
Why are homeomorphism groups important?
For a compact metric space $X$ let $\mathcal H(X)$ denote the set of homeomorphisms in the compact-open topology (also generated by sup metric). It is known that $\mathcal H(X)$ is a Polish ...
3
votes
0answers
140 views
Separating a countable closed set from a point
Let 𝑋 be a separable metrizable space. An intersection of clopen subsets of $X$ is called a C-set.
Question. If each singleton of $X$ is a C-set, $A\subseteq X$ is closed and countable, $x\in X\...
6
votes
1answer
408 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 ...
4
votes
1answer
188 views
One-dimensional compacta as projective limits
Let $X$ be a (not necessarily metrizable) Hausdorff compact space of covering dimension = 1.
Is it possible to express $X$ as a filtering projective limit of finite graphs?
Here finite graphs ...
3
votes
0answers
95 views
Dimension of Alexandrov space which is homeomorphic to a manifold
Let $M^n$ be a smooth manifold of dimension $n$. Let $M$ given a metric with curvature bounded below in the sense of Alexandrov which induces the original topology of $M$.
It is true that the ...
0
votes
1answer
437 views
Dimension of a topological space equals the supremum of the dimension of open subsets in an open cover
For a topological space $X$ which is covered by a family of open subsets $\{U_i\}$, show that $\dim(X)=\sup (\dim(U_i))$.
I understand that $\dim(X)\geq \sup(\dim(U_i))$, so it only suffices to show ...
2
votes
0answers
78 views
Examples of essentially sub-linear functions
A dimension function is an increasing, continuous function $%
f:\mathbb R_{+}\rightarrow \mathbb R_{+}$ such that $f(r)\to 0$ as $r\to 0$.
Say that a dimension function $f$ is essentially sub-linear ...
0
votes
1answer
73 views
Hausdorff outer measure is finite if $\sum_{j = 1}^m |f(x_i)-f(x_{i-1})|^s \le c$ [closed]
Let $f:[0,1] \to \mathbb{R}, G = graph(f)$.
If $\sum_{j = 1}^m |f(x_i)-f(x_{i-1})|^s \le c$ for all partitions $0 = x_0< \ldots < x_m = 1 $ then $H^s(G) < \infty$
What technique can I use to ...
3
votes
0answers
258 views
Covering lemmas in Hochman's ''On self-similar sets with overlaps and inverse theorems for entropy''
I am confused about the covering lemmas in the captioned work and really hope to get some ideas here.
Firstly it is lemma 3.7. (Image of Lemma 3.7) (for convenience here is the lemma of this lemma (...
6
votes
2answers
198 views
Box dimension of the graph of an increasing function
This Hausdorff dimension of the graph of an increasing function shows that:
Let $f$ be a continuous, strictly increasing function from $[0,1]$ to
itself with $f(0)=0, f(1)=1$. Then $dim_H \; G = ...
22
votes
3answers
746 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 ...
3
votes
2answers
484 views
Topological dimension of the image of continuous surjective functions
Consider two topological spaces $(X,\tau)$ and $(Y,\omega)$ and a continuous surjective function $f\colon X\to Y$.
Let $\mathrm{dim}(X)$ and $\mathrm{dim}(Y)$ denote the Lebesgue covering dimension ...
4
votes
1answer
183 views
Graded Grothendieck Group and Hilbert Polynomial
I was wondering if any of the arguments from elementary dimension theory of local noetherian rings could be simplified with knowledge of the Grothendieck group.
Let $A$ be a noetherian graded $K$-...
2
votes
1answer
175 views
Real rank 0 implies stable rank 1 on $C^\ast$-algebras?
A $C^\ast$ algebra has defined stable rank (https://www.univie.ac.at/nuhag-php/bibtex/open_files/2079_Rieffel-StableRank.pdf) and real rank (https://core.ac.uk/download/pdf/82123484.pdf), which are ...
0
votes
0answers
70 views
Set with modified lower box counting dimension strictly less than Hausdorff dimension
Please, can someone give (as simple as possible) example of the set for which modified lower box counting dimension is strictly smaller than Hausdorff dimension?
1
vote
0answers
118 views
Infinite Noetherian ring of dimension $1$ in which distinct non-zero ideals have distinct and finite index
Let $R$ be an infinite commutative ring with unity such that every non-zero ideal has finite index. Then $R$ is Noetherian, every non-zero prime ideal is maximal , and I can also show that $R$ is an ...
5
votes
1answer
391 views
A question on dominant morphism of affine schemes
Let $A \subseteq B$ be a ring extension where $A,B$ are both finitely generated $\mathbb C$-domain of the same Krull dimension. Also assume $A$ is regular (i.e. $A_{ \mathfrak p}$ is regular local ...
32
votes
1answer
906 views
If $\text{dim}(X \times X) = 2\text{dim}(X)$, does $\text{dim}(X^n) = n\text{dim}(X)$?
I have been learning some (topological) dimension theory and have gotten through most of the basic material, at this point, and am about to start looking at papers. In particular, I want to get ...
1
vote
1answer
119 views
Understanding equivalent condition for covering dimension
Let dim $X$ denote the Lebesgue covering dimension for a topological space $X$. Now a result in common books concerning dimension theory states the following:
If $X$ is a normal topological space, ...
