# Questions tagged [dimension-theory]

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. Notice that this tag is not intended for questions about dimension in linear algebra.

460 questions
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### Must a subset of $\mathbb{R} ^n$, that has a dimension of $1$, be formed by a union of lines?

Let $S$ be a subset of $\mathbb{R} ^n$ with a box-dimension of $1$. Is $S$ necessarily a union of lines (including "curved" lines and line segments)? If no, then is $S$ at least an infinite set of ...
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### Covering dimension of boundary of compact subset of $\mathbb{R}^n$

Let $X$ be a compact subset of $\mathbb{R}^n$, with the inherited Euclidean topology. Does it follow that $\dim_{cov}(\partial X)\leq n-1$? I would be happy to have a reference for that, if it is ...
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### The methods to create fractal objects mathematically [closed]

I have read that fractals does not have a mathematical definition as for now. So how do we understand a given object can be considered as a fractal? I know iterated function systems theory is a method ...
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### About homogeneous dimensions of a Lie group

I am looking for a clear and simplified definition of the homogeneous dimension of a vector space or a Lie group. If anyone has the patience to explain me clearly his definition and if possible an ...
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### Small Inductive Dimension is a topological invariant

I have started to read Engelking's book "Theory of Dimesions: Finite and Infinite" and the author states (pp. $3$) that: The small inductive dimension (also called Menger-Urysohn dimension) is a ...
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### Does the Johnson–Lindenstrauss lemma require the Normal distribution?

The Johnson–Lindenstrauss lemma states that a set of points in a high-dimensional space can be embedded into a space of much lower dimension in such a way that distances between the points are nearly ...
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### The box dimension of the graph of a continuous function is $\ge 1$

In the book Interpolation and Approximation with Splines and Fractals by Prof. Massopoust one studies the following theorem with regard two the box dimension of affine fractal interpolation functions: ...
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### Dimension of product of varieties

I've got this exercise asking me to prove first that the product of quasi projective varieties $X$ and $Y$ (henceforth just "varieties") is irreducible iff both $X$ and $Y$ are. I managed to solve ...
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### Is it possible to have a Hausdorff dimension less than the topological dimension?

"Normal" geometric shapes have Hausdorff dimensions equal to their topological dimensions. Mandelbrot defined fractals as shapes that have a Hausdorff dimension greater than their topological ...
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### Dimension image of morphism of projective varieties

Let $f: \mathbb{P}^n \to \mathbb{P}^m$ be a rational map. Then there exists $U \subset \mathbb{P}^n$ open so that $f_{|U}$ is a morphism. What can we say about the dimension of $\overline{f(U)}$? We ...
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