In general topology, dimension theory studies various notions of dimension defined for topological spaces, for example Lebesgue covering dimension, small and large inductive dimension or Hausdorff dimension. In commutative algebra, dimension can be defined for commutative rings.

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A short proof for $\dim(R[T])=\dim(R)+1$?

If $R$ is a commutative ring, it is easy to prove $\dim(R[T]) \geq \dim(R)+1$. For noetherian $R$, we have equality. Every proof I'm aware of uses quite a bit of commutative algebra and non-trivial ...
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Distance and Coordinates in fractional dimensions and the creation of functions with non-integral numbers of paramters.

Background: The Euclidean distance between two points in $n$ dimensions, where $n$ is a positive integer, and position can be described by a vector is given by... $$D_E=\left(\sum_{k=1}^n ...
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36 views

What would a tesseract actually look like?

Say our world is actually 4-dimensional, and I have a tesseract sitting on my table. To a being only capable of perceiving 3 dimensions, what would it actually look like? Would it just look like a ...
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Can these characterisations of finite dimensionality be proven equivalent without using a basis?

I was wondering about how to define "finite dimensional" without talking about bases. Two possibilities occurred to me: Say $V$ is finite dimensional if the canonical inclusion $V\hookrightarrow ...
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207 views

The unit square has dimension two?

Show that the Lebesgue Covering Dimension of the unit square $I^2$ with $I=[0,1]$ is two. I know that a compact subset of Euclidean space $\Bbb R^n$ has Lebesgue dimension at most $n$. So it ...
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142 views

Topological dimension of a countable dense set

I'm reading a (dynamical systems) paper in which topological dimension figures. In my situation, I'm trying to compute the topological dimension of the subset of the $d$-dimensional torus consisting ...
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63 views

What is the point of view when we say $\mathbb{R}^n$ has dimension $n$ in mathematical analysis?

I know there are several different ways to define the dimension. What is the correct point of view to say that $\mathbb{R}^n$ has dimension $n$ in mathematical analysis?
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44 views

Reference Needed: Krull Dimension equals Minkowski Dimension

I'm looking for a reference that basically states that the Krull dimension of an algebraic set in $\mathbb{R}$ equals the Minkowski dimension of the set. It is common knowledge that the Krull ...
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91 views

Defining the Rank of a Projective Module

I am trying to understand the definition of rank for a projective module over a noncommutative ring. The definition I am using is: A sufficient condition for the rank of a free module over a ring ...
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201 views

Canonical $\pi$ dimensional space?

Can we talk about a canonical space of dimension $\pi$? Is there anything like $\mathbb R^\pi$? Have anyone met any fractal of dimension $\pi$?
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19 views

On the covering dimension of an image under a continuous function

I'm trying to solve the following exercise: Let $X$ be a compact Hausdorff space and let $U_1,...,U_n$ be a cover of $X$ of order $m$. Let $z_1,...,z_n\in\mathbb{R}^N$ for some $N$ be in general ...
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49 views

Do mathematicians visualise what 4D would look like when they are working with abstract 4D concepts?

I was wondering if mathematicians do truly have a sense of visualising what the fourth dimension would look like in general, such as the fourth dimension as found in hypercubes. I know that we, as 3D ...
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12 views

Cohomological dimension of an orbit space in Alexander-Spanier theory

Take a ring $R$ and define $cd_R(X)$ ("cohomological dimension over $R$") to be the maximum of the integers $m$ such that $H^m(X; R)\neq 0$, where $H^*$ denotes the Alexander-Spanier cohomology. Now ...
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46 views

Proof of an inequality about set in four-dimensional space

It is needed to prove that for set $A \subset N^4$ $3\log|A|\le \log|\pi_{1,2,3}(A)|+\log|\pi_{1,2,4}(A)|+\log|\pi_{1,3,4}(A)|+\log|\pi_{2,3,4}(A)|$ where $\pi_{i,j,k}(A)$ is a projection of $A$ on ...
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74 views

Prove that this condition is true on an Zariski open set

Why is there an Zariski open set of $P\in GL(n,\mathbb{R})$ such that $P\,\text{diag}(1,\dots,1,-1)P^{-1}$ can be conjugated by a diagonal matrix $D$ to get an orthogonal matrix? Note that ...
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43 views

prove the subspace is equal to L1∩L2

Let $L1,L2$ be subspaces of $V$ and $dim(L1+L2)=1+L1 \cap L2$ Show that $L1 \subseteq L2$ or $L2 \subseteq L1$ and $L1+L2= L1$ or $L2$ it's has something to do with if $L1,L2 \ne L1 \cap L2$ ...
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46 views

Hausdorff dimension of a ball

Let $\{f_1,\dots,f_m\}$ be an IFs and $E_n$ be the associated self similar set. It's known that $E_n$ is a union of disjoint balls $B(x_i,R\cdot r^n)$ (balls with same radius but not the same ...
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36 views

Hausdorff dimension of closed intervals is not changed under $f(x)$

Let $f(x)=x^2$. Prove that for any $E\subseteq\mathbb{R}$ the dimension of image is not changed i.e $$\dim_HE=\dim_H(f(E))$$ Any set in $\mathbb{R}$ can be represnted as a countable union of ...
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101 views

Von Neumann and Hausdorff continuous dimensions are related?

Von Neumann in his book Continuous Geometry introduced (in a suitable lattice) a dimension function that has a continuous range. The definition of a dimension function is axiomatic: see Continuous ...
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45 views

Depth and kernel of multiplication map

If $A$ is a commutative $k$-algebra, $\mu: A\otimes_k A\rightarrow A$ is its multplication map and $I$ is the kernel of that map (viewed as an $A\otimes_k A$-module map) then what is the relationship ...
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76 views

Krull dimension of localization

If $R$ is a commutative ring and $m$ a maximal ideal therein, then what are the conditions for the Krull dimension of $R$ equaling to the Krull dimension of $R_m$?
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57 views

Hochschild dimension

I'm curious; if $A$ ia a commutative $k$-algebra over a field $k$ of global dimension $n$, then is its $A^e$-projective dimension $2n$ (this is also sometimes called the Hochschild cohomological ...
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21 views

On the definition of (small) inductive dimension

A regular topological space $X$ has inductive dimension smaller or equal to n if and only if: ($n=-1$) $X=\emptyset$; ($n>-1$) The space $X$ has a base of opens $\mathscr{B}$ such that, for all ...
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38 views

Number of touching triplets in hypersphere packing contact graph

I would like to know what is curently known about the number $N$ of touching-triplets involving a given vertex $V$ in the contact-graph of $d$-hypersphere packings. For $d=2$ and $d=3$, one has $N = ...
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15 views

Can correlation dimension of an attractor exceed the dimension of the space?

Here is the definition of the correlation dimension: http://en.wikipedia.org/wiki/Correlation_dimension Is there a proof that the correlation dimension cannon exceed the dimension of the space?
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70 views

Kernel of an infinite dimensional operator

Can the Kernel of an infinite dimensional operator have $\dim=0$? I am thinking to the annihilation ($\hat E$) and creation ($\hat E^\dagger$) operators. Suppose, in fact, we have an infinite but ...
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33 views

How can i evaluate a parabolic region in 2-Dimensions onto n-Dimensions?

I am working on the K-Nearest Neighbor algorithm and have the solution to a problem as a parabola in 2-dimensional space. Should I continue this solution onwards to n-dimensions (and thereby create ...
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65 views

Buckingham Π-Theorem

I'm about to conduct some experiments concerning a welding process. To prepare for this I wanted to do a dimensional analysis of the process. So I read a lot about the Π-Theorem and I was able to ...
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66 views

Separation of Euclidean Space

Consider a finite collection $\mathcal{H}$ of hyperplanes of $\mathbb{R}^n$ that have a common line. Given some $A \subseteq \mathbb{R}^n$ that is homeomorphic to a subset of $\bigcup\mathcal{H}$, ...
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178 views

Hilbert (polynomial) dimension and dimension of a support of a module

$\newcommand{\Supp}{\mathrm{Supp}}$ $\newcommand{\Ann}{\mathrm{Ann}}$ Let $X$ be an affine algebraic variety (over a field $K$, can assume it is algebraicaly closed), $M$ a finitely generated ...
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42 views

Covering dimension of the union

Let $\{{A_n}\}$ be the closed subsets of $X$, such that ${A_n} \subset \operatorname{Int}{A_{n + 1}}$ and $ \cup {A_n} = X$, if $A_1$ and all $\operatorname{cl}(A_n-A_{n-1})$ have the covering ...
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56 views

Disjointness of stars in a simplicial complex in $\ell_2$

Definitions Let's consider a full $n$-dimensional simplicial complex $C$ in $\ell_2(X)$. By that I mean a set of all functions $f:X \to[0,1]$ such that $\sum_{x\in X}f(x)=1$ and there are at most ...
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14 views

Hyperplane Equation

Would it be correct to say that the function $$ A\cdot T_1+B\cdot T_2+C\cdot T_3=D\cdot x_1+E\cdot x_2+F\cdot x_3+G $$ Generates a hyperplane in 6D? (A through G are constant parameters) Thank you.
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What is the dimension of the space $W_2$ if $\text{dim}(W_1+W_2) = 2$, $\text{dim}(W_1 \cap W_2) = 1$ and $W_1 = \langle a_1, a_2 \rangle$?

What is the dimension of the space $W_2$ if $\text{dim}(W_1+W_2) = 2$, $\text{dim}(W_1 \cap W_2) = 1$ and $W_1 = \langle a_1, a_2 \rangle$, where $a_1 = (1,0,0)$, $a_2=(2,0,0)$? Okay so I know and ...
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20 views

Can we estimate the dimension of some function space?

Can we estimate the dimension of some function space? e.g. if $X$ is a locally compact Hausdorff space, what is the (linear) dimension of $C(X)$(as a linear space), which represents the set of all ...
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18 views

Higher dimension leads to different optimization?

It appears z= f(x,y) has a global max/min at another particular (x,y). Using only one independent variable $x$ at fixed $y$ i.e., for z = f(x) I get another max/min point for $x$ optimum point ...
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56 views

Example of affine variety with infinite dimension

It is a typical question in Algebraic Geometry to find a Noetherian topological space with infinite dimension: we can take $X=[0,1]$ and check that. However, it is not so obvious for me to: Find ...
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41 views

Length of a vector in 3 dimension spaces

while I'm studying principal-component analysis, I have misunderstanding about principal component, I was think that it's orthogonal and has a single point,which I mean by single points is if we have ...
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Compute the dimesion of these subsets of $\mathbb{R}^4$

I need to compute the codimension of the subsets $S_1,S_2,S_3\subseteq\mathbb{R}^4$ given by \begin{align} S_1&=\{(a,b,c,d)\in\mathbb{R}^4:ad-bc\neq0,\},\\ ...
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16 views

Infimum of fractal dimensions left by almost-disjoint-disk covering of plane

Answers to this question show that it is not possible to cover the plane by almost-disjoint closed disks. As note in the comments, the Appollonian Gasket leaves a set of fractal dimension ...
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To prove existence of an open set of functions

I am trying to prove the following: In $C(X,Y)$ with $X=[0,1]$ and $Y$ of finite dimension $K$, $C(X,Y)$ having the topology of uniform convergence, for any $K$ finite there exists an open set of ...
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Vertex Figure of Leech Lattice

I'm doing some research on the newton number (kissing #) problem and it would be extremely useful to know the type and number of elements, particularly the facets, of the vertex figure formed from the ...
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Identity of dimensions related to cohomology group of projective space

Let $k$ be a field, $S=k[T_0,\ldots T_r]$, $\mathbb P=\mathbb P_k^r=\operatorname{Proj}S$ and $O$ a structure sheaf of $\mathbb P$. How can I show the identity $$\operatorname{dim}_kH^0(\mathbb ...
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19 views

Extensions to higher dimensions by tensorization. Unitary DFT in 2D?

I have problem understanding the underlying concept of tensoration (if there is such term). Fist of all the unitary DFT is NxN. Is it 1D ? How does it look when we increase the dimension let say to ...
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61 views

Find a basis and state its dimension of a $C$-vector space polynomial.

The $C$ vector space $V$ of polynomials $P(t) \in C[t]$ of degree at most $n$ and such that $P(a) = P'(a) = 0$ for $a \in C$ fixed. Indication : prove that $P(t) \in V \Leftrightarrow (t − a)^2$ ...
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76 views

prove $\dim\mathbb{Z}[X_1,X_2]=3$ from first principles

Since $\dim R[X] =\dim R+1$ for any Noetherian ring $R$, the ring $\mathbb{Z}[X_1,X_2]$ must have dimension 3. But how can this be proved 'from first principles', i.e. without using any big theorems ...
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$\dim V + \dim (V^{\perp} + W) = \dim W + \dim (V + W^{\perp}).$

Show that if $V$ and $W$ are two subspaces of $\Bbb{R}^n$ it is true that $\dim V + \dim (V^{\perp} + W) = \dim W + \dim (V + W^{\perp}).$ I used some properties like $\dim (V^{\perp} + W) = \dim ...
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54 views

Covering dimension of a compact metric space

I would like to see the proof of the following fact (references appreciated). A compact metric space $X$ has covering dimension $\leqslant n$ if and only if there is a continuous surjection $\pi ...
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54 views

Examples and counterexamples in dimension theory

I am looking for examples of topological spaces that are interesting from a dimension theoretical point of view - in particular I am looking for these three notions of dimension: Little inductive ...
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29 views

Equal fields and dimension

Let $K \subseteq L$ be fields. Show that K=L if and only if $dim_k L=1$ I know that I will have to show both directions of the implication. I'm very new to the topic of fields so I'm still ...