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).
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Planar compact connected set whose boundary has a finite length is arcwise connected
Let $K \subset \mathbb{R}^{2}$ be a compact connected set such that $\mathcal{H}^{1}(\partial K)<+\infty$. Is $K$ arcwise connected?
user524824
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The number of components of the preimage of a continuum for a polynomial
Given a polynomial f with degree d, we can define a dynamical system $(\mathbb{C}, f)$. If we have a proper continuum $N \subset \mathbb{C}$, it is known that the set $f^{-1}(N)$ has at most $d$ ...
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Why is this continuum circle-like?
A continuum is a compact connected metrizable space.
A continuum $X$ is called arc-like if for every $\varepsilon>0$ there is an open cover $U_1,\ldots,U_n$ of $X$ such that the diameter of $U_i$ ...
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Is the Mandelbrot set Suslinian?
The Mandelbrot set is known to be (path-)connected and compact. A non-degenerate space with these properties is called a continuum.
A continuum $X$ is Suslinian if every collection of non-degenerate ...
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Does the pseudo-arc contain Erdős space?
The pseudo-arc is the unique hereditarily indecomposable chainable continuum.
The Lelek fan is the unique compact, connected subset of the Cantor fan (the cone over the Cantor set) with a dense ...
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Find at least one square-boxed subcontinuum
Recall that a plane continuum is a closed, bounded,
connected subset of the plane.
It is non-degenerate if it contains at least two points.
(We may sometimes just say "continuum" even if
we ...
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Are there hereditarily square-boxed plane continua?
A plane continuum is a bounded, closed and connected subset of the plane.
A bounding box $B$ for a plane continuum $C$ is
a rectangle $B=[a,b]\times[c,d]$ (including sides and interior)
such that $C$ ...
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How “disconnected” can a continuum be?
A continuum is a compact connected metrizable topological space.
Given a cardinal $\kappa$, a topological space $X$ is called $\kappa$-connected if it is not possible to write $X$ as the disjoint ...
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Irreducible subcontinuum without Zorn's lemma
In continuum theory we frequently use the fact that two points in a continuum are contained in an irreducible subcontinuum.
A continuum $X$ is a compact connected metric space. A subcontinuum $K\...
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$n$-connected spaces (terminology)
A graph is called $n$-connected if it remains connected after removal $\le n$ vertices.
Question. What is the name of an analogous property of topological spaces: a space that remains connected after ...
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For which continua $X$ does $X^\mathbb{N}$ have the fixed-point property?
A topological space $X$ has the fixed-point property if any continuous map $f : X \to X$ has a fixed point (i.e., an $x$ such that $f(x) = x$). It's well known that certain topological spaces, such as ...
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Is a circle of circles necessarily a 2-manifold?
Let $X$ be a continuum (a compact connected metric space).
Suppose that there is a collection of simple closed curves $\mathcal C$ which partitions $X$ into pairwise-disjoint, nowhere dense sets. ...
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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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Is there an uncountable family of "hereditarily unembeddable" continua?
Define a family $\{C_i\}_{i\in I}$ of continua, that is compact connected metrizable spaces, to be hereditarily unembeddable (a name I just made up) iff for all $i\neq j$ no nontrivial subcontinuum of ...
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Two maps into $[0,1]$ are equal at some point
In the paper below, there appears the following theorem:
whose proof is left to the reader. It's not immediately obvious how I would prove this. How about the special case $X=Y=[0,1]$? It seems to be ...
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When does $C(X)$, $X$ a continuum, admit a continuous choice function?
Given a continuum $X$ (compact metrizable connected $X$) let $K(X)$ denote the hyperspace of nonempty compact subspaces of $X$ with the Vietoris topology and let $C(X)$ denote the (closed) subspace of ...
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Isotopy Classes and Embeddability of Products in $\mathbb{R}^2$
On MSE I asked if the plane contains an uncountable collection of mutually disjoint copies of the Warsaw Circle; it seems to be false, and is probably already known but I'm not sure that anybody has ...
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(Seeking Definition) What Does it Mean for a Space to have Rim-Type $\alpha$? Or the 'derivative' of a Countable Set?
I've encountered a definition in several papers, but literally none of them define the term. They all instead reference a book by Menger that has never been printed in English. The term is "rim-...
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Shrinkable homogeneous compact and connected $T_2$-space
A topological space $(X,\tau)$ is said to be homogeneous if for any $x,y\in X$ there is a homeomorphism $\varphi:X\to X$ such that $\varphi(x) = y$. Moreover, we say that $(X,\tau)$ is shrinkable if ...
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Isotopy Classes of Non-Connected Planar Sets
I was just looking through some of my old questions on StackExchange and noticed that this one went totally unresolved. I still think it'd be useful as a lemma for plane topology if it were true, and ...
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Can an "almost injective'' function exist between compact connected metric spaces?
Let $\pi: X\to Y$ be a surjective continuous function between the compact, metric and connected spaces $X$, $Y$, and $Y_0 = \{y\in Y: \#\pi^{-1}(y)>1\}$. Suppose that:
$Y_0$ is dense in $Y$,
$Y\...
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Which compact metrizable spaces have continuous choice functions for non-empty closed sets?
Let $X$ be a compact metrizable space and let $\mathcal{K}_{ne}(X)$ be the collection of non-empty closed subsets of $X$ with the Vietoris topology (i.e. the topology induced by the Hausdorff metric ...
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Is every locally compact connected homogeneous metric space a manifold cross a continuum?
Suppose that $(X,d)$ is a locally compact connected homogeneous metric space, where by homogeneous I mean that for any $x_0,x_1 \in X$ there exists an isometry $f:X\rightarrow X$ such that $f(x_0)=x_1$...
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Rough classification of Peano Curves
By Peano curve I mean a continuous map from the unit interval that fills the unit square in $\mathbb R^2$.
In the paper:
Shchepin, E. V.; Bauman, K. E., Minimal Peano curve, Proc. Steklov Inst. Math....
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Polish transversals
A subset of $X$ an indecomposable continuum $Y$ is called a composant transversal if $X$ has exactly one point from each composant of $Y$.
So a continuum has a composant transversal precisely when ...
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A continuum which is both Suslinean and non-Suslinean?
Continuum means compact connected metrizable with more than one point.
A continuum is Suslinean if every collection of pairwise disjoint subcontinua is countable.
There is an apparent contradiction ...
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Planar arc on a topologically embedded sphere or disk in $\mathbb{R}^3$
An arc is a set homeomorphic to the unit interval $[0,1]$; an arc in $\mathbb{R}^3$ is planar if it is contained in some plane.
The following questions are motivated by Anton Petrunin's Disc bounded ...
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Do solenoids embed into Möbius strips?
I found a strange attractor which looks a lot like a solenoid. The attractor continuum is the closure of a continuous line which limits onto itself, and it is locally homeomorphic to Cantor set times ...
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Topological Shape Operator More Sensitive than Inverse Limits
This is a very general sort of question, and the use of the phrase 'shape operator' is a bit sloppy since there is already an established "shape theory." But what I have are topological spaces that ...
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Is each Peano continuum a topological fractal?
Problem. Is each Peano continuum a topological fractal?
A compact Hausdorff space $X$ is a topological fractal if $X=\bigcup_{i=1}^n f_i(X)$ for some continuous maps $f_1,\dots,f_n:X\to X$ such that ...
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Does every cut-point space embed into the plane?
Let $X$ be a connected separable metrizable topological space. Call it a cut-point space if $X\setminus \{x\}$ is disconnected for every $x\in X$. Then does $X$ embed into the plane?
My thoughts:
(...
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Does each separator between points of a continuum contain an irreducible separator?
Definition. A closed subset $S$ of a topological space $X$ is called a separator between points $x,y\in X\setminus S$ if the points $x$ and $y$ belong to different connected components of $X\setminus ...
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Limit of homeomorphisms from square to square
Let $\square=[0,1]\times[0,1]$ be the unit square
and $f\colon\square\to \square$ is a continuous map that fixes the points on the boundary.
Assume $f$ is a limit of homeomorphisms $\square\to \...
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Is each metric continuum $\ell_p$-chain connected?
This problem was motivated by the MO problems:
"Running most of the time in a connected set", "Is every metric continuum almost path connected?" and "Are $\varepsilon$-connected components dense?".
...
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Are $\varepsilon$-connected components dense?
Let $X$ be a connected compact metric space. Given a positive $\varepsilon$ and two points $x,y\in X$ we write $x\sim_\varepsilon y$ if there exists a sequence $C_1,\dots,C_n$ of connected subsets of ...
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Is every metric continuum almost path-connected?
The question was motivated by this question of Anton Petrunin.
By a metric continuum we understand a connected compact metric space.
Let $p$ be a positive real number. A metric continuum $X$ is ...
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Running most of the time in a connected set
Let $P$ be a compact connected set in the plane and $x,y\in P$.
Is it always possible to connect $x$ to $y$ by a path $\gamma$ such that the length of $\gamma\backslash P$ is arbitrary small?
...
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Example of a non-locally connected continuum
Continuum $=$ compact connected metric space.
Let $X$ be a continuum. $X$ is indecomposable means that every proper subcontinuum of $X$ is nowhere dense in $X$.
It is easy to see that if $X$ is ...
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A minimal continuum
A continuum $X$ is called minimal if it is not a single point and is homeomorphic to all its nontrivial subcontinua.
Here a trivial continuum is a single point.
What is an example of a minimal ...
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Disc bounded by a plane curve
Let $\Sigma$ be a sphere topologically embedded into $\mathbb{R}^3$.
Is it always possible to find a disc $\Delta\subset\Sigma$ which is bounded by a plane curve?
It is easy to find an open disc ...
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