# Questions tagged [continuum-theory]

For questions from continuum theory. A continuum is a compact connected metric space (sometimes this term is used for a compact connected Hausdorff space). Do not use this tag for questions related to the Continuum Hypothesis in Set Theory.

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### Prove that an inverse limit of arcs, as well as an inverse limit of simple closed curves cannot contain a simple triod.

I found this question in chapter 2 of Nadler's Introduction to Continuum theory but I'm a bit lost on how to prove this. I've already searched this question and I found that an inverse limit of arcs ...
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### Is the pseudoarc nowhere locally connected on any subspace which does not contain isolated points?

Let $(X, \mathcal{T})$ be the pseudoarc, which is a hereditarily indecomposable continuum. Here hereditary means on every subcontinuum. Subcontinuums of a continuum are exactly its closed and ...
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### Tensor Product Rules

I've just started a new term of uni and I just started a module in continuum mechanics, this being said there's some things that seem pretty important that I'm new to. The first exercise in the notes ...
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### Mixed product of second-order tensors and vectors

I was studying the angular momentum equation in the continuum case and I encountered this identity. I am not sure how the identity is derived. Could some one supply more details and intermediate step? ...
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### Tensors In Continuum Mechanics And Artificial Intelligence

I learned about tensors in a math course in grad school. It was about the scalars, vectors, and higher-order tensors used in physics and differential geometry. It talked about metric tensors, co-...
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### Divergence of tensor fields

I have found numerous definitions for the divergence of a tensor which makes me confused as to trust which one to use. In Itskov's Tensor Algebra and Tensor Analysis for Engineers, he begins with ...
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### Is $X \times [0, 1)$ a linear continuum? What about $X \times (0, 1]$?

I'm reading from Topology by Munkres and in example 2 of section 24, 'Connected Subspaces of the Real Line', the author discusses that $X \times [0, 1)$ is a linear continuum in the dictionary order ...
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### Is the following set a continuum?

Let F be a non-empty family of subcontinua of a continuum X such that for any finite subfamily $F_{1},F_{2},...,F_{n}$ in F there is $C\in F$ such that $C \subset F_{1} \cap F_{2} \cap... \cap F_{n}$ ...
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### Help with the proof of the following theorem

I have the following theorem and its proof but I can't understand some steps of the proof I hope you can help me. If X is a chainable continuum and $C = \{U_{1}, . . . , U_{n}\}$ is a ε–chain in $X$ ...
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### Can a solenoid exist in the plane?

If $X$ is an $n$-dimensional continuum, then $X$ can be embedded in $\mathbb{R}^{2n+1}$. So if $X$ is a solenoid, it can be embedded in $\mathbb{R}^3$, we even have a construction of this. Is it ...
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### How to linearly order the set of all subsets of real numbers?

I wondered if there are linearly ordered sets of any cardinality. As I understand it, there are. But I want to see at least one concrete example of a linearly ordered set which cardinality is greater ...
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### Examples of continua that are contractible but are not locally connected at any point

A continuum is a compact, connected, metrizable space. What are examples of continua that are contractible but nowhere locally connected, meaning that no point has a neighbourhood basis consisting of ...
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### Prove that $X$ doesn't have cut points

Let $X$ be a metric continuum such that for every two points $a,b \in X$ the set $X\setminus \{a,b \}$ isn't connected. Prove that $X$ doesn't have cut points. First I tried to prove it by ...
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### How strong is Sierpiński theorem about continua?

I've just learned about a theorem by Sierpiński, that a continuum can't be partitioned into countably many non-empty closed sets. Can we partition some continuum into $\aleph_1$ non-empty closed sets ...
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### Where can I find more insight about spaces of subsets of a base space?

I've been studying Michael's article "Topologies on spaces of subsets" and he states some propositions and lemmas without proving, asserting that they follow directly from the definitions ...
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### Is there a classification of all possible types of continua?

A continuum is a compact connected metric space. The continuum $X$ is called a Peano continuum if it is locally connected. A chain in the topological space $X$ is a collection $U_1,U_2,\ldots ,U_n$ of ...
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### How to prove that hausdorff metrics on a continuum generate the same topology?

We define a continuum $X$ as a compact, connected and nonempty metric space and we denote the closed hyperspace as $2^{X} = \{A \subset X: A\neq \emptyset \text{ is closed }\}$. If $X$ is a continuum, ...
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### If $f$ is atomic then int $(f (U)) \neq \emptyset$ (interior). True or False?

Let $f:X \to Y$ be a continuous function between continua. If $f$ is atomic then int $(f (U)) \neq \emptyset$ (interior). I don't know if this conjecture is true. Before presenting my attempt, I ...
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