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Questions tagged [dimension-theory]

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

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Dimension of the graph of a function $\varphi : \mathbb R^2 \to \mathbb R$

Let $\varphi : \mathbb{R}^2 \to \mathbb{R}$ be a continuous function, and let $G(\varphi)$ be the graph of $\varphi$. Denote $R:=\{(x,0) \in \mathbb{R}^2 | x \in \mathbb{R}\}$ as the real line in $\...
Chicken feed's user avatar
3 votes
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What is the Lebesgue covering dimension of this topological space?

Take the 4 dimensional time-oriented spacetime $(M,g)$ such that it's not strongly causal. Take the induced topology defined by the Lorentzian metric called Alexandrov topology. This topology matches ...
Bastam Tajik's user avatar
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1 answer
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Box dimension and graph of Hölder function

In Kamont "ON THE FRACTIONAL ANISOTROPIC WIENER FIELD" (found here : https://www.math.uni.wroc.pl/~pms/files/16.1/Article/16.1.6.pdf), on page 96, it is claimed that, if a function $f:I^{d}\...
BabaUtah's user avatar
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Existence of finite 3-dimensional hyperbolic balanced geometry

Together with @TarasBanakh we faced the problem described in the title. Let me start with definitions. A linear space is a pair $(S,\mathcal L)$ consisting of a set $S$ and a family $\mathcal L$ of ...
Ihromant's user avatar
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Conformal welding and Jordan loop consequences?

In the similar context as Conformal welding of rectifiable curves In classical conformal welding theory, we start with a homeomorphism $h$ of the unit circle and try to find a Jordan domain $D$ ...
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Local dimension of stationary measures for iterated function systems with an expanding map

Consider the iterated function system (IFS) $X_n$ on $I = [0,1] $generated by the functions $\Phi = \{f_1,f_2,f_3\}$ and the probability vector $P = (p/2,p/2,1-p),$ where: $f_1,f_2: I\to I$, where $...
Matheus Manzatto's user avatar
2 votes
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How far can one get by counting spaces of solutions this way?

I am quite used to "counting"/computing finite dimensions. For example, one would expect a hypersurface in $\mathbb{C}^3$ to have dimension $3 - 1 = 2$. But it is often the case that the ...
Malkoun's user avatar
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Dimension of a subspace of $n\times n$ real symmetric matrices

Let $n\in \mathbb N.$ Let $W$ be a non-trivial subspace of $n\times n$ symmetric matrices such that for every $x\in \mathbb R^n\setminus \{0\}$ there exists $a_x\in \mathbb R^n\setminus \{0\}$ such ...
mathew's user avatar
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$\sigma$-product of the Hilbert cube

Given a homogeneous space $X$ and $p\in X$, we define the sigma product to be the following subspace of $X^\omega$: $$\sigma X=\{\mathbf x \in X^\omega:x_n=p\text{ eventually}\}$$ ("eventually&...
D.S. Lipham's user avatar
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Is every weakly $1$-dimensional space embeddable in the plane?

A $1$-dimensional (separable metric) space $X$ is weakly $1$-dimensional if $$\Lambda(X)=\{x\in X:X\text{ is 1-dimensional at }x\}$$ is zero-dimensional (i.e. the space $\Lambda(X)$ has a basis of ...
D.S. Lipham's user avatar
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Does the pseudo-arc contain Erdős space?

The pseudo-arc is the unique hereditarily indecomposable chainable continuum. The Lelek fan is the unique compact, connected subset of the Cantor fan (the cone over the Cantor set) with a dense ...
D.S. Lipham's user avatar
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7 votes
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Can you remove a zero dimensional subspace from a cube and obtain a planar space?

The question, which came up in a conversation with my advisor Ola Kwiatkowska, is pretty much in the title: Let $Z\subseteq[0,1]^3$ be zero-dimensional. Is it possible for $[0,1]^3\setminus Z$ to be ...
Alessandro Codenotti's user avatar
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Iterating the dimensional kernel of a metric space

Fix $n\in \mathbb N$. Let $X$ be a separable metric space of (inductive) dimension $n$. Let \begin{align} \Lambda(X)&=\{x\in X:X\text{ is $n$-dimensional at }x\}\\ \\ \Lambda^2(X)&=\Lambda(\...
D.S. Lipham's user avatar
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The real dimension of any real algebraic set equals the complex dimension of its complexification

I want to prove the following statement. Please help! Given any semialgebraic set $A$, consider its real Zariski closure $V_{\mathbb{R}}$ (which always has the same real dimension of $A$). Now ...
user86954's user avatar
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A generalized Hausdorff dimension in form of a Lower semi continuous function

Let $(X,d)$ be a compact metric space. Assume that $f:X\to \mathbb{R}$ is a positive continuous function. We say that the $f$-dimension of $(X,d)$ is equal to $0$ if for every $\epsilon>0$ ...
Ali Taghavi's user avatar
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1 answer
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Seeking for references - Bowen Formula and a link between dimension theory and thermodynamic formalism

I'm needing references - preferably published papers and books - about this subject. I'm relatively new to the state of the art of fractal geometry and am way too inexperienced to seek for myself at ...
anchova's user avatar
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2 answers
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Why do almost all points in the unit interval have Kolmogorov complexity 1?

Re-posted from math.stackexchange as I did not get any answers there. I am reading Jin-yi Cai, Juris Hartmanis, On Hausdorff and topological dimensions of the Kolmogorov complexity of the real line, ...
i like math's user avatar
3 votes
1 answer
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Dimension of sumset vs sum of dimensions

Let $A\subset \mathbb R$. Is it true that $$ \dim(A+A)\le 2\dim A $$ for some dimensions – say, lower box for the LHS and upper box for the RHS.
Nikita Sidorov's user avatar
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Commutative Banach algebras with zero-dimensional maximal ideal space and disjoint supports of Gelfand transforms

Let $A$ be a commutative semi-simple unital Banach algebra and let $\Delta$ be the maximal ideal space of $A$. Denote by $\widehat{\cdot}\colon A\to C(\Delta)$ the Gelfand transform. If $\Delta$ is ...
user491354's user avatar
6 votes
1 answer
405 views

Does finite Hausdorff dimension imply finite packing dimension?

In other words, does there exist a metric space $(E,\rho)$ with finite Hausdorff dimension but infinite packing dimension? Here are my thoughts: I know that it is generally hard to relate Hausdorff ...
Peter Koepernik's user avatar
4 votes
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Dimension properties of some concrete hereditarily disconnected subspaces of the Hilbert cube

This question was motivated by this MO-question asking about the example of a hereditarily disconnected metrizable separable space, which is not the union of countably many totally disconnected ...
Taras Banakh's user avatar
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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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6 votes
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Classification of Polish spaces up to a $\sigma$-homeomorphism

A function $f:X\to Y$ between topological spaces is called $\bullet$ $\sigma$-continuous if there exists a countable cover $\mathcal C$ of $X$ such that for every $C\in\mathcal C$ the restriction $f{\...
Taras Banakh's user avatar
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10 votes
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Is there a universal totally disconnected Polish space?

A topological space $X$ is called 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 $y\notin U$. In 1973 Roman Pol proved that ...
Taras Banakh's user avatar
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1 vote
1 answer
165 views

Dimension-preserving non-linear map

Let $F:\mathbb{R}^n\to\mathbb{R}^n$ be a continuous non-linear map, and let $A$ be a connected subset of $\mathbb{R}^n$ with $\text{dim}(A)=d\leq n$. When can we say that the dimension of the image, $\...
RS-Coop's user avatar
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Sufficient conditions for the covering dimension and large inductive dimension of compact Hausdorff spaces to coincide

I have been looking through Alan Pears' "Dimension theory of general spaces" recently. In this book Pears references a 1960 paper by Aleksandrov and Ponomarev called "Some classes of $n$...
Robert Thingum's user avatar
2 votes
1 answer
357 views

Fraction dimensional "Euclidean" space

The “dimension” of Euclidean space $\mathbb{E}^n$ can be explained as an algebraic property, simply as a dimension of a vector space over the field $\mathbb{R}$. It also can be understood as a ...
Kacper Kurowski's user avatar
38 votes
1 answer
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Sequences with 0's in $\mathbb R ^\omega$

Let $\mathbb R ^\omega$ be the set of all sequences of real numbers in the product topology. Let $X$ be the set of all sequences in $\mathbb R ^\omega$ which have at least one 0. Let $Y$ be the set of ...
D.S. Lipham's user avatar
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2 votes
0 answers
156 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 ...
Jörg Neunhäuserer's user avatar
3 votes
1 answer
240 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 ...
Nik Bren's user avatar
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0 answers
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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 ...
D.S. Lipham's user avatar
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2 votes
0 answers
367 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 ...
Roxana's user avatar
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3 votes
0 answers
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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 ...
John Samples's user avatar
2 votes
1 answer
120 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 ...
ABIM's user avatar
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6 votes
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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 ...
D.S. Lipham's user avatar
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6 votes
1 answer
147 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 ...
Thomas Yang's user avatar
0 votes
1 answer
97 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. ...
Learnmore's user avatar
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0 answers
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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 ...
Taras Banakh's user avatar
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1 vote
0 answers
287 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,...
Loli's user avatar
  • 111
5 votes
1 answer
282 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
  • 4,501
6 votes
2 answers
313 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, ...
Somatic Custard's user avatar
9 votes
2 answers
503 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 ...
D.S. Lipham's user avatar
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0 votes
0 answers
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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 $V_{j}(t)=V_{j}(t,\xi_{j})$...
demolishka's user avatar
4 votes
1 answer
200 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 ...
usercsw's user avatar
  • 41
9 votes
2 answers
577 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
28 votes
2 answers
2k 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
1 answer
131 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 ...
D.S. Lipham's user avatar
  • 3,105
1 vote
1 answer
230 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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5 votes
0 answers
158 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 ...
Manlio's user avatar
  • 332
4 votes
0 answers
437 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 $n<\omega$}\}...
D.S. Lipham's user avatar
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