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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Derivation of the Beltrami–Michell equation
I would like to derive Beltrami–Michell equation. I consider linearised elasticity theory, and assume that the material of interest is an isotropic elastic solid with the constitutive relation
$$\tau =...
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The number of components of preimage of a continuum under a polynomial
Given a ploynomial $f$ with degree $d$, then we have a dynamical system $(\mathbb{C},f)$, the critical points are those points $z$ such that the derivative of $f$ at which is zero. If we have a ...
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Adding point to connected open set
Let $X$ be a compact, connected, locally connected space. Let $U$ be a connected open subset of $X$. Let $p\in \overline U$. Clearly $U\cup \{p\}$ is connected.
Is $U\cup \{p\}$ locally connected?
Is $...
user410185
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Every cutting of a continuum introduces two noncut points.
I'm reading the section on continua in Willard's General Topology. His Lemma 28.7 reads
If $K$ is a continuum and $(p,U,V)$ is a cutting of $K$, then each of $U$ and $V$ contains a noncut point of $K$...
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Metric space that can be written as the finite union of connected subsets but isn't locally connected
I'm looking for an example of a metric space $X$ such that for every $\epsilon > 0$ there exist connected subsets $A_1, \dots A_n$ for some $n \in \mathbb{N}$ such that $X = \cup_{i = 1}^nA_i$ and ...
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Proving a Sierpiński result on partitions of the unit interval into closed sets
From Are the Sierpiński cardinal ˊn and its measure modification ˊm equal...?, I seem to have rediscovered a result from Sierpinski:
Theorem (Sierpiński, 1921). For any countable partition of the ...
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How is a second material derivative defined?
I am reading a paper and it used a second material derivative written like this:
$$
\dfrac{D^2\delta}{Dt^2}
$$
I know the first order material derivative operator is defined
$$
\dfrac{D\delta}{Dt}=\...
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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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Is an indecomposable continuum nowhere locally connected?
Let $(X, \mathcal{T})$ be an indecomposable continuum. A continuum is a compact connected metric space. A continuum is indecomposable if it is not a union of two proper subcontinuums.
Is it true that $...
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Cardinality of the set of measure zero sets
I have been thinking about this question for a while now and found nothing on the matter so far.
Assuming the continuum hypothesis (or maybe also for the case that we assume that it is false), what is ...
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On compact connected components in the complex plane.
If $x\in \mathbb{C}$ and $r>0$, denote by $B(x,r)$ the open ball in $\mathbb{C}$ with center $x$ and radius $r$.
Suppose that $A\subset B(0, \rho)$ is compact, and that $A_{0}$ is a connected ...
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Bounding boxes of subcontinua
Let $C$ be a closed and bounded subset of the plane.
A bounding box $B$ for $C$ will by definition be the smallest rectangle with vertical and horizontal sides that contains $C.$
(We allow rectangles ...
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Why is the Sierpinski carpet connected and locally connected?
The Wikipedia article on the Sierpinski carpet fractal says that it is compact, connected and locally connected.
It is clear from the construction that the Sierpinski carpet is closed and bounded in $\...
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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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set limits, lim sup and lim inf
Let $X$ be continuum and $\{C_i\}$ a sequence of compacts set in $X$ then $\limsup C_i$ and $\liminf C_i$ is compacts.
where
$(C_i)_{i=1}^{\infty} \subseteq \mathcal{P}(X)$ and
$\liminf C_i =\{x \in ...
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Suppose $Y \subset X$, both are continua of dimension one. If $\check{H}^1(X;\mathbb{Z}) = 0$, is $\check{H}^1(Y;\mathbb{Z}) = 0$?
Suppose $X$ is a continuum (a compact connected Hausdorff space, not necessarily metrizable) of dimension one and $Y$ is a subcontinuum of $X$ (i.e. a subspace of $X$ which is a continuum). If the ...
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(0,2)-Tensor associated Matrix
I am studying continuum mechanics with an introduction of tensor calculus. First of all I wanna say that this is my very first time i see tensor calculus, so I have a lot of things that are not clear ...
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Measure zero sets on Peano Spaces.
Let $P$ be a Peano space. Recall that $P$ is a Hausdorff space that is a continuous surjective image of $[0,1]$.
The standard Peano curve $f:[0,1]\to [0,1]^2$ is self-intersecting and the set $\{x\in[...
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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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Snake-like continuum vs Peano continuum
A snake-like continuum is a continuum such that for every $\varepsilon >0$ there exist a collection of open sets $d_1,d_2,\ldots d_n$ with diameters les than $\varepsilon$ such that $d_i \cap d_j\...
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Is there a topological characterization of Euclidean spaces?
Suppose $X$ is a topological space.
What are the properties such that if $X$ satisfies them, then $X$ is homeomorphic to $\mathbb{R}^{n}$ for some non-negative integer $n$?
There are answers to this ...
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How does being an indecomposable continuum imply irreducibility between all pairs of points?
A continuum $K$ is called indecomposable if $K$ can not be written as the union of two proper subcontinua $A,B$.
A continuum $K$ is called irreducible between points $x,y\in K$ if there is no proper ...
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Sets closed in generalized arcs.
I have been reading the following article. I have a question in Lemma 2.3 about the closed sets $\mathcal{A}$ and $\mathcal{B}$ that are presented.In summary, my question is the following:
A ...
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Can we classify continua of $\mathbb{R}^2$ transformed under homeomorphsisms and bi-measurable maps?
Definition. If there exists a sequence of Borel measurable sets $X_1, ..., X_n\subseteq \mathbb{R}^2$ with positive Lebesgue measure and maps $f_i:X_i\to X_{i+1}$ such that each $f_i$ is either a ...
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Cardinality of the set of arguments in integration
Consider a function $f(x)$ with domain $D_f=\{x\in \mathbb{R} \space|\space 0\le x\le1 \}$
Since $D_f$ is an interval of $\mathbb{R}$, it is an uncountable set and its cardinality is $\aleph_1$ under ...
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Brouwer Reduction Theorem
I went looking for a statement of the Brouwer Reduction Theorem, but Google only gives hits for his fixed point theorem. I talked to an old professor of mine about it being used to prove if you have ...
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Two time derivatives of kinetic energy of fluid
Suppose $D$ is a smooth domain, $\rho > 0$ is fluid density (constant) and $u \in C^1([0,1];D)$ is the fluid velocity. Let $K(t) = \frac{1}{2}\int_D \rho \vert u \vert^2 dV,~0\le t \le 1,$ be the ...
user171342
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Blockers in hyperspaces. $\mathcal{B}(C_{\infty})=\mathcal{B}(2^X))$
Let $X$ be a locally connected continuum. Then
$$\mathcal{B}(C_{\infty}(X))=\mathcal{B}(2^X)$$
This is theorem 1.5 of the article "Alejandro Illanes, Paweł Krupski, Blockers in hyperspaces, ...
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Local Contractibility of Homogeneous, Locally Path-Connected Spaces
I was wondering about the following:
Is it possible for a compact, metric, topological group to
be locally path-connected but not locally contractible?
Here "locally contractible" means ...
- 303
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Pathological Examples in Smash Products?
I don't normally encounter smash products in the stuff I do, but rn I'm looking at some point-set properties of what are apparently smash products. My impression is that smash products usually arise ...
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Can Two Compact, Non-Separating Subsets of the Plane be Separated by Jordan Curves?
Suppose $X, Y \subset \mathbb{R}^2$ are compact, mutually disjoint, and neither separates the plane.
How does one prove that there are disjoint Jordan curves $J_1, J_2$ such that $X$ is in the ...
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Product of Mobius Band with Itself
I have a few questions about the Mobius band $M$. The first two questions are pretty direct, whereas the third is a bit vague.
Let $F_2(M)$ denote the collection of non-empty subsets of $M$ with at ...
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Finite-Dimensional Homogeneous Contractible Spaces
Suppose that $X \subset \mathbb{R}^n$ is compact, homogeneous and contractible (and thus connected). Does $X$ have to be a point?
I couldn't think of a non-trivial example, and there isn't a ...
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Naming a property of continua
Let me define a continuum $X$ to be nice if for each pair $a,b\in X$ of distinct points, there are subcontinua $M,N\subseteq X$ such that: $a\in M\setminus N$; $b\in N\setminus M$; and $M\cup N=X$. (...
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Does $\mathbb{R}^2$ Contain Uncountably Many Disjoint Copies of the Warsaw Circle?
The Warsaw Circle is defined as the closed topologist's sine curve, with an additional arc attached at its free end point and one of the end points of the critical line:
Since we don't have an ...
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