# Tagged Questions

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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### How do I derive the Minkowski dimension of the following set?

I have been trying to (rigorously) get the Minkowski's dimension (refer here for a basic definition) of a parabola: $(x,y) \in \mathbb{R}^2 : \{ y = x^2\}$.
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### Hausdorff dimension of a Cantor Set: attaining a lower bound

I'm considering the problem of calculating the Hausdorff dimension of a Cantor set, according to the following lemmas: Lemma 1 Let $C: [0, 1] \rightarrow [0, 1]$ be a Cantor staircase function. Then ...
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### Let $U\subset\Bbb{R}^n$ be open bounded nonempty. What is the topological dimension of $\partial U$?

An exercise in Munkres' Topology (Exercise 50.8) shows that if $X$ is a $\sigma$-compact Hausdorff(*) space s.t. every compact subset has topological dimension $\leq n$, then so does $X$. If we define ...
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### Hausdorff dimension via ergodic theory

This is definitely a soft question, but it was recently mentioned to me that one can study the dimension of fractals via ergodic methods. I'm familiar with ergodic theory on about the level of ...
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### Lower bound on dimension of fibres of a dominant mophism of irreducible affine varieties

Whilst doing exercise $11.4.B$ of Ravi Vakil's "Foundations of Algebraic Geometry", I got stuck with the following problem (although I think that many of the hypotheses are unnecessary and a more ...
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### Dimention of span of subspace

Let $V$ be a vector space over$\mathbb R$ and $B = { v_1,v_2,v_3 }$ a base of $V$. $S = { v1-v2, v2-v3, v3+v1 }$ $A$. is $S$ a base of $V$? $B$. is $dim(span(s)) = 1$ ? I really hope I correctly ...
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### Dimension Theory

Let $F$ be a closed subset of a manifold $M$ of dimension $n$ such that $M\setminus F$ is disconnected and F contains no open sets. Is that true the small inductive dimension of F is $n-1$? Small ...
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### Why the topological dimension of C is 2?

From what I know, the topological dimension of a set has to do with open sets covering it, homeomorphic to R^{n}. Then we can cover C with balls, for instance,of <...
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### Doubts in Volume, Hypervolume in $R^4$

Recently I was reading about triple integrals and I came across the statement - "We saw that a double integral could be thought of as the volume under a two-dimensional surface. It turns out ...
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### Is a hypersphere of non-integer dimension a fractal?

Thanks to the gamma function the formula for the surface of a unit http://mathworld.wolfram.com/Hypersphere.html $$S(n) = \frac{2 \pi^{n/2}}{\Gamma(n/2)}$$ allows to calculate the surface of a ...
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### Can a polygon be one dimensional?

When looking up the definition of polygon, Wikipedia tells me: In elementary geometry, a polygon /ˈpɒlɪɡɒn/ is a plane figure that is bounded by a finite chain of straight line segments closing ...
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### Lie Algebra: Dimension of a one-parameter group (dimension of orbit)?

I am reading a book about Applications of Lie Groups to Differential Equations by Peter Olver. Lets say we have a PDE with $p$ independent variables and $q$ dependent variables In Chapter 3.1 (page ...
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### Hausdorff Dimension of a Smooth Manifold

I read a book about fractals which stated without proof: Every $m$-dimensional $(m<n)$ smooth manifold $M$ in $\mathbb{R}^n$ has Hausdorff dimension $m$. How can we prove this?
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### When the self-similar dimension and the Hausdorff dimension are different?

By the en.wikipedia, for the self-similar sets in a metric space, the self-similar dimension and the Hausdorff dimension are often the same, but not always. Is there a known sufficient-necessary ...
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### Simulating 4D on a 3D mean

Why if we can simulate 3D on a 2D mean, why isn't possible to simulate 4D on a 3D mean (real world)? Has someone tried?
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### Orthogonal planes in n-dimensions

In three dimensions two planes are orthogonal when their normal vectors are orthogonal (their inner product is zero). For example, planes $xy$ and $xz$ are orthogonal because the vectors $\hat{z}$ and ...
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### How can I understand is the picture $2D$ or $3D$

I can not understand is this picture 2D or 3D. What is the rule or condition to be a 2D or 3D picture. How can I understand that? Please help me!
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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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### Polish groups having finite covering dimension

The paradigm examples of Polish groups that have finite covering dimension are obviously $\mathbb{R}^n$ for any finite $n$. They are also locally compact. Is it possible to construct a sequence $G_n$ ...
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### Are there any practical applications for higher dimensional geometry?

My current understanding of higher dimensions is that we can take geometric properties from 2D or 3D, and simply extend them further theoretically. So in the same way that 3D objects can be projected ...
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### Dimensionality of shapes in two space

I'm having trouble properly understanding the dimension of shapes. Take a square or triangle. Are the boundaries of these two-dimensional shapes one-dimensional? Is the entire figure two dimensional? ...
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### Why study dimensions?

I am quite new to the forum so please feel free to correct/give pointers if I am posting something in the wrong place. I have been perusing "Dimension Theory" by Witold Hurewicz & Henry Wallman ...
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### Dimension of a Noetherian topological spaces

We know that the definition of (Krull) dimension of a Noetherian topological spaces $X$ is the following:  \dim X=\max\{n\in\mathbb{N}\mid\emptyset=Z_{-1}\subsetneqq Z_0\subsetneqq\dots\subsetneqq ...
### Proving rank$(S\circ T)\le$ rank $S$ and null$(S\circ T)\ge$ null $T$ [closed]
Let $T$:$V\rightarrow W$ and $S$:$W\rightarrow U$ be linear transformations. I need to prove the two statements in the title. I don't know how to approach this problem. I am quite sure it involves ...