# 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 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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### 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 ...