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I suggest to create a tag synonym analysis-and-odes $\to$ ca.classical-analysis-and-odes.
The tag analysis-and-odes has only five questions and empty tag info - so it does not seem to have any distinction from ca.classical-analysis-and-odes. Four of the five questions in this tag were asked by t...
2
I'd like to propose creation of continuum-theory tag.
To me (as an outsider, but still a bit interested in this topic) it seems that continuum theory is an area of general topology which enjoys some interest both among topologists and among mathematicians in general. (For example, one part of Op...
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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?
Comments:
Be aware of pseudoarc --- it is a compact connected set which contains no nontrivial p...
6
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 diameter $<\varepsilon$ in $X$ such that $x\in C_1$, $y\in C_n$ and $C_i\cap C_{i+1}\ne\emptyset...
3
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?".
Let $p$ be a positive real number. A metric space $(X,d)$ is called $\ell_p$-chain connected if fo...
7
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 indecomposable then every connected open subset of $X$ is dense in $X$.
Question. Are these two cond...
3
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 $X\setminus S$. A separator $S$ is called an irreducible separator between $x$ and $y$ is $S$ coincides with ...
12
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 $\square\to \square$. (By Moore's theorem it is equivalent to the condition that for any point $p\in \square$ the i...
9
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 called $\ell_p$-almost path-connected if for any points $x,y\in X$ and any $\varepsilon>0$ here exist...
11
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 which boundary lies in a plane,
but the boundary might be crazy; for example it might be Polish ...
4
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:
(0) It is not difficult to see that $X$ must have dimension $1$, and therefore embeds into $\mathb...
2
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 have trivial shape in the classic Borsuk sense (and thus also in the sense of Cech homology). How...
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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 for any sequence $(g_i)_{i\in\omega}\subset\{f_1,\dots,f_n\}^{\omega}$ the intersection $\big...
