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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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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26
votes
1
answer
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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 ...
13
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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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answer
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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 ...
4
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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 ...
2
votes
0
answers
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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 $C$ ...
10
votes
2
answers
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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 ...
10
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1
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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 ...
6
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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 ...
4
votes
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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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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?".
...