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).

Filter by
Sorted by
Tagged with
29 votes
1 answer
800 views

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? ...
Anton Petrunin's user avatar
26 votes
1 answer
820 views

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 ...
Anton Petrunin's user avatar
13 votes
0 answers
252 views

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 ...
Wlodek Kuperberg's user avatar
11 votes
1 answer
326 views

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 ...
TopologicalDynamitard's user avatar
4 votes
0 answers
112 views

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 ...
Mirko's user avatar
  • 1,345
2 votes
0 answers
156 views

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 $C$ ...
Mirko's user avatar
  • 1,345
10 votes
2 answers
248 views

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 ...
James Hanson's user avatar
  • 10.3k
10 votes
1 answer
508 views

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 ...
Taras Banakh's user avatar
  • 40.8k
6 votes
1 answer
164 views

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 ...
Taras Banakh's user avatar
  • 40.8k
4 votes
0 answers
97 views

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 $f(x_0)=x_1$...
James Hanson's user avatar
  • 10.3k
4 votes
0 answers
208 views

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?". ...
Taras Banakh's user avatar
  • 40.8k