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Questions tagged [continuum-theory]

Continuum Theory is the study of nonempty, connected, compact, metric spaces, referred to as Continua. In some more general contexts a continuum is any nonempty, connected, compact, Hausdorff space. Herein lies the study of spaces such as the Sierpinski Carpet, the Warsaw Circle, and the Topologists' Sine Curve as well as various fixed point theorems. Do not use this tag for questions related to the Continuum Hypothesis in Set Theory.

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Trace of diffusion equation matrix

I'm studying waves moving through mediums and I come across this type of equation a lot while describing acoustic waves speed and stress on the material, $$\begin{gather} \frac{\partial}{\partial z}\...
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Velocity gradient in polar coordinate

I just want to convert velocity gradient in polar coordinate to velocity gradient in Cartesian one (i.e. $\frac{\partial u_r}{\partial r}=f\left(\frac{\partial v}{\partial y}\right)$). How can I ...
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The components of velocity field are given, determine the components of velocity gradient tensor, deformation tensor and rotation tensor

The components of velocity field are given, v1 = cyz ; v2 = cxz ; v3 = 0 a) determine the component of the velocity gradient tensor b) determine the component of the deformation tensor c) ...
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Finite Unions of Dendrites

I will ask the main question first, and then give the motivation for this one! The question is a bit specific, but seems to be the most general question to ask after handling some obvious ...
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Spaces That Have Uncountably Many Disjoint Copies in $\mathbb{R}^2$

There is a theorem by Moore that says there are not uncountably many disjoint copies of the simple triod in the plane (the simple triod is the space by adjoining one end point from three copies of $[0,...
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Distinguishing Two Compactifications of $[0,1)$

Pictured below are two subsets of the plane, each a compactification of the closed half-line with remainder a closed arc. I am really frustrated by my inability to prove that the space pictured on ...
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Existence of Connected Sets in Complements of Closed Sets

Suppose that we have a connected $T_1$ space $X$ and two disjoint, closed subsets $A, B$. Then if $X$ is also locally connected, it is a theorem that there is a connected subset $C \subset X \...
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Arcs Contained In Continuous Injections of $[0,1)$

Suppose we have a metric space $X$ and a continuous injection from $[0,1)$ onto $X$. The case I had in mind will satisfy that $X$ is compact, but the problem I have can be stated more generally, as ...
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Factorization of Products of Manifolds

This is a fairly general question but I couldn't find a source that deals with this problem in the sense I was wondering. Suppose that we are given two topological spaces $X, Y$ such that $X \times Y ...
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Homology/Cohomology Theories for Non-Locally Connected Spaces

This is a very general question. I am wondering what homology or cohomology theories have been developed for the analysis of spaces which are not locally connected. Particularly, theories that ...
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How to Show that $X \times \mathbb{R}$ is not Homeomorphic to $\mathbb{R}^2$

This is a geometrically obvious statement, but I only just realized that an explicit proof might not be entirely trivial. I was curious to know if anybody knows an elementary proof of this fact, and ...
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The $n$-dimension continuum as an ordered set

It is well known that we can define the linear continuum as a linear order $(S, \le)$ with no first or last element which is complete and an enumerable subet $B$ exists dense in $S$. Indeed, one ...
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Best Software for Generating Images (Topology)

I would like to know what people would recommend as the best program to use for generating nice images, particularly of continua. For example, what's the best program available that makes it easy to ...
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Can a countable dense subset be split into two disjoint dense subsets?

Suppose $X$ is a compact connected Hausdorff space and $D \subset X$ countable and dense. Can we always write $D=D_1 \cup D_2$ as a disjoint union of countable dense subsets? More generally if $U \...
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Equivalence of Path-Connectedness and Arc-Connectedness for Hausdorff Spaces

I have a classical sort of question. If we define a path in a space $X$ from points $a$ to $b$ to be a continuous image $f: I \rightarrow X$ from the unit interval to $X$ such that $f(0) = a$ and $f(...
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Prob. 6, Sec. 24, in Munkres' TOPOLOGY, 2nd ed: For a well-ordered set $X$, $X\times[0,1)$ in the dictionary order is a linear continuum [duplicate]

Here is Prob. 6, Sec. 24, in the book Topology by James R. Munkres, 2nd edition: Show that if $X$ is a well-ordered set, then $X \times [0, 1)$ in the dictionary order is a linear continuum. ...
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Non-metric topological continua

What important results hold for non-metric continua, or where can I find a survey of such results? There are three definitions of a continuum around: a non-empty topological space that is (1) ...
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Local Bases in Locally Connected and Semi-Locally Connected Spaces

I was trying to come up with a counterexample for something and kept failing and failing, so I was hoping someone could help me out. The setting is compact, connected, metric spaces (continua) so ...
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Conditions Weaker than Locally Euclidean

This is a very general question, but hopefully some people find it interesting. I'm working in the setting of compact metric spaces, so most of the basic topological properties will be satisfied. ...
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A Local Connectedness Condition for Compact, Connected Metric Spaces

I am having trouble proving a result from a paper, which of course includes no proof. I wonder if the author had a simple - but flawed - argument in mind, or if I'm just being a dunce. It is Theorem ...
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Looking for Sierpinski Carpet Universality Reference

I am currently working through Sam Nadler's "Continuum Theory" and have become stuck on an exercise (1.16) which says "Prove that the Sierpinski Carpet is universal." By universal they means that the ...
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Characterisation of One Dimensional Peano Continua

Question: Is every one dimensional peano continuum a countable union of arcs which pairwise intersect only in their end points? Definitions: Continuum - A connected, compact metric space Peano ...
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Covering Dimension of a Subset of the Plane

I am proving the equivalence of a few statements, and I'm stuck on one of them. I want to show that: If $K \subset [0,1]\times[0,1]$ is a non-degenerate continuum with empty interior, then $K$ has ...
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Every Subcontinuum of a $1$ Dimensional Locally Path Connected Plane Continuum is Locally Path Connected

EDIT The question was originally stated in a general setting, but now it is in terms of plane continua as these are the cases I care about. A $continuum$ is a compact, connected metric space. By a $...
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Pathological Continua which are Path Connected and Locally Path Connected.

I'm doing research in generalised inverse limits, and I'm trying to prove a result about circle-like plane continua. Definitions A continuum is a compact, connected metric space. A plane continuum ...
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Equivalence relation on a continuum

Let $X$ be a continuum $=$ a connected compact metric space. Define $x\sim y$ if $x$ and $y$ are contained in a nowhere dense subcontinuum of $X$. It is easy to see that $\sim$ is an equivalence ...
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Separating continuum in the disc

Continuum = compact connected set. Suppose that $U$ and $V$ are nonempty disjoint open subsets of $[0,1]^2$. Is there necessarily a continuum $K\subseteq [0,1]^2$ that divides $U$ and $V$? More ...