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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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 =...
David Mayer's user avatar
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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 ...
Yee Neil's user avatar
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1 answer
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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 $...
user avatar
2 votes
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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 $K$...
Ningxin's user avatar
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4 votes
1 answer
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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 ...
H4z3's user avatar
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1 answer
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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 ...
Hew Wolff's user avatar
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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}=\...
Logan's user avatar
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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 ...
kaba's user avatar
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2 votes
1 answer
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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 $...
kaba's user avatar
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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 ...
Staub und Dreck's user avatar
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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 ...
user 987's user avatar
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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 ...
Mirko's user avatar
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How to solve the equation to plot the deformation curve of the actuator?

Thanks for reading this. I ask the question from the following research article (https://pubs.rsc.org/en/content/articlelanding/2019/sm/c9sm01672d#:~:text=When%20the%20magnetic%20field%20strength,...
Vignesh Sahadevan's user avatar
1 vote
1 answer
162 views

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 $\...
jenda358's user avatar
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lim sup of a sequence of sets and $(lim\ sup\ A_n)_{\varepsilon}$

I'm trying to prove that given a sequence of nonempty sets $(A_n)_{n\in \mathbb{N}}$ of a compact, connected and metric space $X$ and $\varepsilon > 0$ there exists $M \in \mathbb{N}$ such that for ...
Iñaki Mendieta's user avatar
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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 ...
murpw2011's user avatar
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1 answer
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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? ...
crostata's user avatar
2 votes
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121 views

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-...
duffymo's user avatar
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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 ...
Bjaam's user avatar
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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 ...
codehumor's user avatar
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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 $C \subset F_{1} \cap F_{2} \cap... \cap F_{n}$ ...
Urixml23's user avatar
2 votes
0 answers
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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 $X$ ...
M98's user avatar
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1 answer
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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 ...
Jakobian's user avatar
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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 ...
ZFC abuser's user avatar
7 votes
1 answer
126 views

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 ...
TopologicalDynamitard's user avatar
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0 answers
61 views

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 ...
mathMaría's user avatar
3 votes
1 answer
57 views

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 ...
jl00's user avatar
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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 ...
cento18's user avatar
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0 answers
51 views

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[...
O-Schmo's user avatar
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1 vote
1 answer
82 views

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 ...
Ryoiki Tenkai's user avatar
5 votes
1 answer
187 views

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 ...
Jakobian's user avatar
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0 votes
1 answer
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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 ...
Bryant's user avatar
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2 votes
0 answers
79 views

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 ...
Emo's user avatar
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0 answers
51 views

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, ...
Urixml23's user avatar
3 votes
0 answers
38 views

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 ...
user1999's user avatar
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2 votes
1 answer
75 views

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\...
Emo's user avatar
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3 votes
3 answers
328 views

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 ...
Maximal Ideal's user avatar
2 votes
2 answers
116 views

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 ...
Robert Thingum's user avatar
0 votes
1 answer
113 views

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 ...
Haus's user avatar
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2 votes
0 answers
17 views

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 ...
Jakobian's user avatar
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0 votes
1 answer
87 views

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 ...
chicken_game's user avatar
3 votes
1 answer
121 views

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 ...
Tom Kacvinsky's user avatar
1 vote
1 answer
125 views

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 ...
user avatar
2 votes
0 answers
28 views

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, ...
user1999's user avatar
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2 votes
0 answers
108 views

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 ...
John Samples's user avatar
1 vote
0 answers
75 views

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 ...
John Samples's user avatar
3 votes
1 answer
96 views

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 ...
John Samples's user avatar
2 votes
0 answers
73 views

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 ...
John Samples's user avatar
20 votes
0 answers
315 views

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 ...
John Samples's user avatar
1 vote
1 answer
52 views

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 $M\cup N=X$. (...
jinpa's user avatar
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