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.
31
questions
1
vote
1answer
16 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
$$\mathrm{dev}(u_{i,j})=u_{i,j}-\frac{1}{...
1
vote
0answers
20 views
Direct proof that the space of endpoints of the Lelek fan is almost zero-dimensional
I'm interested in the proof that space of endpoints of the Lelek fan is almost zero-dimensional. As much is claimed in On homogeneous totally disconnected 1-dimensional spaces by Kawamura, Oversteegen,...
0
votes
0answers
29 views
Best resources about Prime End Theory
I'm trying to find the best source for understanding prime ends of planar continua, in particular, of planar continua that do not separate the plane. What I want to know is how one can make use of ...
1
vote
2answers
64 views
Measurability of $\frak c$
Can the continuum cardinality $\frak c$ be a measurable cardinal ?
Why not, what is the simplest reason besides that ZFC proves existence of $\frak c$ but not existence of a measurable cardinal?
3
votes
0answers
22 views
Continua with Type 4 Prime Ends
For those who may feel they need the definitions in prime end theory, I provided what I think is sufficient below. A lot of what's below comes from a paper by John C. Mayer, "Inequivalent Embeddings ...
0
votes
1answer
42 views
Using Alexander lemma to prove that if $X$ is continuum then the hyperspace $2^X$ is compact
So far in the books I've read all proofs involving Alexander Lemma to prove Tychonoff's theorem or in general any
compact space they use Zorn Lemma argument. So I was wondering if is there any ...
1
vote
0answers
40 views
Trace of diffusion equation matrix
I'm studying waves moving through mediums and I come across this type of equation a lot while describing acoustic waves speed and stress on the material,
$$\begin{gather}
\frac{\partial}{\partial z}\...
1
vote
0answers
70 views
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 ...
1
vote
0answers
71 views
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 ...
7
votes
0answers
113 views
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,...
4
votes
1answer
189 views
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 ...
0
votes
0answers
109 views
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 \...
3
votes
0answers
60 views
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 ...
3
votes
1answer
74 views
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 ...
5
votes
1answer
170 views
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 ...
3
votes
2answers
179 views
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 ...
3
votes
0answers
51 views
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 ...
28
votes
1answer
372 views
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 \...
5
votes
1answer
267 views
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(...
1
vote
1answer
272 views
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.
...
5
votes
2answers
130 views
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) ...
2
votes
1answer
74 views
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 ...
6
votes
1answer
226 views
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.
...
4
votes
2answers
200 views
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 ...
2
votes
1answer
109 views
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 ...
2
votes
2answers
84 views
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 ...
2
votes
1answer
85 views
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 ...
3
votes
1answer
93 views
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 $...
2
votes
1answer
137 views
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 ...
12
votes
2answers
325 views
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 ...
1
vote
1answer
45 views
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 ...
