# 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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### 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 $\mathcal M$ a Borel subset of $C(X)$?

Let $X$ be a continuum (compact connected metric space) and for each $x\in X$ let $$M_x=\{y\in X:\exists \text{ a nowhere dense subcontinuum of }X \text{ containing }x\text{ and }y\}.$$
Suppose that $...

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### Is every atriodic nonseparating plane continuum chainable?

There exist atriodic tree-like continua which are not chainable, though the examples that I'm aware of are very complicated.
Is every atriodic tree-like plane continuum chainable? This is equivalent ...

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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 ...