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.
109 questions
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Is $A_{\epsilon}=\{x \in X: d(x,A) \leq \epsilon\}$ a continuum metric space?
Let $(X,d)$ a continuum metric space and $A \subset X$ subcontinuum metric space. For all $\epsilon >0$ we define $A_{\epsilon}=\{x \in X: d(x,A) \leq \epsilon\}$. Is $A_{\epsilon}$ a continuum?
We ...
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Order $\omega$ in a dendrite
before I begin, I would like to provide some definitions and theorems.
Definition. A topological space $(X, \tau_X)$ is a continuum, if $X$ is a non-empty, metric, compact and connected space.
Here $\...
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Relation between free arcs and branch points in continua
before I begin, I would like to provide some definitions and theorems.
Definition. A topological space $(X, \tau_X)$ is a continuum, if $X$ is a non-empty, metric, compact and connected space.
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A result about dendrites.
before I begin, I would like to provide some definitions and a theorem.
Definition. A topological space $(X, \tau_X)$ is a continuum, if $X$ is a non-empty, metric, compact and connected space.
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Free arcs in the Universal dendrite
Definition. A topological space $(X, \tau_X)$ is a continuum, if $X$ is a non-empty, metric, compact and connected space.
Definition. Let $X$ be a continuum and define $E(X)=\{p \in X: ord_X(p)=1\}$, $...
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Can we divide an interval $(0,1)$ into countably-infinite number of equal parts? If so, will the first part have real numbers in it?
In the context of surreal numbers, can we divide the real inteval $(0,1)$ into a countably-infinite number of equal parts, for instance, into $\omega$ parts? By "equal" I mean such that one ...
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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 ...
- 1,210
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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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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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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 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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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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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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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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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 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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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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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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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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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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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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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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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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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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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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