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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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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7 votes
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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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3 votes
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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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Examples of local homeomorphisms between continuums

I've been looking for and thinking about examples of local homeomorphisms between continua. I have found two. $f:S^1 \to S^1$ defined by $f(z)=z^n$. $f: \sum_2 \to \sum_2$ defined by $f(\{z_n\}_{n=1}^...
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2 votes
1 answer
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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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3 votes
2 answers
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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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1 answer
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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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1 answer
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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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1 answer
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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 ...
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2 votes
0 answers
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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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2 votes
0 answers
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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 ...
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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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2 votes
1 answer
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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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2 votes
0 answers
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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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18 votes
0 answers
243 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 ...
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1 vote
1 answer
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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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16 votes
1 answer
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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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2 votes
1 answer
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Boundaries in Spaces where Quasicomponents and Components Coincide

Let's call a space $X$ geometric if its components and quasi-components coincide. Let's also define a property called the boundary bumping property: $X$ has the boundary bumping property ("bbp&...
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8 votes
0 answers
298 views

Examples of Finite-Dimensional Space with Non-Vanishing Homology in Higher Dimensions?

The Barratt-Milnor Sphere $X_n$ is an $n$-dimensional space which has non-vanishing singular homology in arbitrarily high dimensions. The space $X_n$ is a generalized Hawaiian Earring, i.e. the $n$-...
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1 vote
2 answers
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Local connection of closure of open connected subspaces in some metric spaces.

Let $X$ be a metric, connected, locally connected and locally compact space, and let $U$ be an open connected subspace of $X$. Is $\overline{U}$ always locally connected? I´m writing a thesis about ...
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2 votes
0 answers
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Question about $3$-Sphere Characterization

Suppose that $X$ is a $3$-dimensional ENR continuum (a compact, connected, metric space that is a Euclidean neighborhood retract). Suppose that $X$ also satisfies the following: If $a, b \in X$ and $...
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6 votes
1 answer
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Any proof (in English) that Generalised Peano Continua are continuous images of the real line?

By generalised Peano continuum I mean a metric, connected, locally connected, locally compact space. I found a paper called ...
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1 vote
2 answers
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Sierpinski Triangle as Finite Union of Dendrites

Can the Sierpinski Triangle be written as a finite union of dendrites? If so, can it also be verified what the minimal number is (assuming you can't do it with just two)? This is a small piece from a ...
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0 votes
0 answers
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Consequence of Hahn - Mazurkiewicz theorem

Definition 1. A continuum is a compact, connected, non-empty metric space. Definition 2. An arc is any space $X$ homeomorphic to the closed interval $[0, 1].$ Definition 3. A continuum of Peano is a ...
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4 votes
1 answer
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Can there be maximal proper subcontinua?

Let X be a compact connected Hausdorff space (a continuum). Say $C$ is a proper subcontinuum: a proper compact connected subspace of $X$. Can $C$ be maximal? That is: is it possible that for all ...
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1 vote
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Every decomposable continuum contains a 2-od.

I have to prove that every decomposable continuum contains a 2-od. These are some of the definitions: A continuum is a nonempty compact, connected and metric space. A continuum $X$ is said to be ...
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3 votes
3 answers
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Density and continuua for sets without linear orders

A dense order is defined as an ordered set such that for any $x$ and $y$ such that $x < y$, there is a $z$ such that $x < z < y$. A linear continuum is defined as a linearly ordered set that ...
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2 votes
1 answer
76 views

Continuum cannot be written as Countable Union of Disjoint Closed Subsets

Continuum: $X$ is a continuum if it is a compact connected Hausdorff space. A continuum $X$ cannot be equal to $\bigcup\limits_{n=1}^\infty F_n$ for any $\{F_n\}$, where $\{F_n\}$ is a collection of ...
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0 votes
1 answer
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Two equivalent definitions of endpoints of continuum

In different materials there are used different definitions of an $\it{endpoint}$ in a continuum (compact and connected metric space). Some of them are equivalent, but some of them have minor ...
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1 vote
1 answer
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Is every hereditarily unicoherent plane continuum non-separating?

Let $X$ be a connected compact subset of the plane. Suppose that $A\cap B$ is connected for every two closed connected subsets $A,B\subseteq X$. Then is $X$ non-separating? That is, is $\mathbb R^...
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0 votes
1 answer
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How to prove this equation about calculation of matrix determinant?

How to prove the equation about the determinant of Matrix $M$, i.e., $|M|=\frac{(M \cdot a) \times (M \cdot b) \cdot (M \cdot c)}{a \times b \cdot c}$ where $a$, $b$ and $c$ are arbitrary vectors. ...
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1 answer
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The set I × I (where × denotes the Cartesian product and I = [0, 1]) in the lexicographic order is a linear continuum.

I've found on wikipedia a proof but I don't really understand ot. If a topological space $S$ (order topology) is linear continuum it satisfies the next: a) $S$ has the least-upper-bound property b)...
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0 votes
1 answer
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Peano Continuum and a set countable of arcs...

Let Y be a Continuum. Then there is a set countable of arcs $$\{l_{n}\phantom{a}|\phantom{a}n\in\mathbb{N}\}$$ such that $$Z=Y\cup{\left(\bigcup_{n\in\mathbb{N}}{l_{n}}\right)}$$ is a Peano ...
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2 votes
0 answers
41 views

Isotopy Classes of Non-Connected Planar Open Sets

I am wondering about a question which I have to assume has been studied, either for its own sake or sporadically in the context of other questions, but I've never come across any papers on the matter. ...
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1 vote
1 answer
127 views

Tensor deviator calculation rules

In the field of continuum/solid mechanics, there are often deviatoric tensors defined, like for the derivation (comma in einstein notation) of a displacement $$\mathrm{dev}(u_{i,j})=u_{i,j}-\frac{1}{...
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