All Questions
Tagged with plane-geometry gn.general-topology
30 questions
1
vote
2
answers
127
views
Homeomorphism and boundary of a complementary component
Let $X\subset \mathbb R^2$ be compact and connected. My question is whether homeomorphisms of $X$ preserve boundaries of complementary components.
More precisely, let $h:X\to X$ be a homeomorphism.
...
- 3,317
6
votes
0
answers
98
views
Rigid plane curves
A curve is a continuous one-to-one image of the real line $\mathbb R$.
A space $X$ is rigid if the only homeomorphism of $X$ onto itself is the identity.
Is there a rigid curve in the plane?
I am ...
- 3,317
2
votes
1
answer
107
views
Mandelbrot boundary and component of $\infty$
Let $M$ be the Mandelbrot set, and $\partial M$ its boundary. So $\partial M$ is the set of those points $z\in M$ such that every neighborhood of $z$ contains a point of $\mathbb R^2\setminus M$.
Let $...
- 3,317
3
votes
1
answer
203
views
Simple closed curves in a simply connected domain
Let $U$ be a bounded simply connected domain in the plane. Let $K$ be the boundary (or frontier) of $U$. For every $\varepsilon>0$ is there a simple closed curve $S\subset U$ such that the ...
- 3,317
6
votes
5
answers
953
views
Two arcs in the complement of a disc must intersect?
Let $D=\{z\in \mathbb C:|z|\leq 1\}$ be the unit disc in the complex plane, with interior $U=\{z\in \mathbb C:|z|<1\}$.
Let $A\subset \mathbb C\setminus U$ be an arc intersecting $D$ only at its ...
- 3,317
3
votes
2
answers
203
views
Recovering a set from its projections in varying coordinate systems - a projection hull?
Let me describe the simplest non-trivial case of what I have in mind. Let $V$ be a 2-dimensional $\mathbb{R}$-vector space and fix an isomorphism $V \cong \mathbb{R}^2$, where $\mathbb{R}^2$ is ...
- 7,127
3
votes
0
answers
69
views
Is every weakly $1$-dimensional space embeddable in the plane?
A $1$-dimensional (separable metric) space $X$ is weakly $1$-dimensional if $$\Lambda(X)=\{x\in X:X\text{ is 1-dimensional at }x\}$$
is zero-dimensional (i.e. the space $\Lambda(X)$ has a basis of ...
- 3,317
2
votes
2
answers
150
views
A plane ray which limits onto itself
A ray is a continuous one-to-one image of the half-line $[0,\infty)$.
If $f:[0,\infty)\to \mathbb R ^2$ is continuous and one-to-one, then we say that the ray $X=f[0,\infty)$ limits onto itself if for ...
- 3,317
25
votes
6
answers
2k
views
Are there infinitely many "generalized triangle vertices"?
Briefly, I'd like to know whether there are infinitely many "generalized triangle centers" which - like the orthocenter - are indistinguishable from a vertex of the original triangle. This ...
- 21.2k
7
votes
0
answers
207
views
Is the derivative the unique operation on points in the plane that preserves convexity?
Let $C(n)$ be the space of multisets of size $n$ of points in the Euclidean plane, topologised appropriately, and consider a surjective continuous map: $$D:C(n)\rightarrow C(n-1)$$
Such that the ...
- 1,949
7
votes
1
answer
183
views
Stability Question for Isotopies Between Compact Sets
Suppose $X, Y$ are compact sets in $\mathbb{R}^2$ and $F$ is an ambient isotopy carrying $X$ onto $Y$.
Is there an ambient isotopy $F'$ agreeing with $F$ on $X$ and which is constant in a ...
- 439
6
votes
1
answer
300
views
Proof of Denjoy-Riesz Theorem and Moore's Generalization?
The Denjoy-Riesz Theorem states that any compact zero-dimensional subset of the plane can be covered by an arc, i.e. an embedded image of $[0,1]$. Sometimes it's stated just for covering a Cantor Set,...
- 439
11
votes
2
answers
305
views
Connecting a compact subset by a simple curve
Let $K$ be a compact subset of $\mathbb R^n$ with $n\ge 2$ (say if you like $n=2$, which is possibly sufficiently representative).
Q: Does there exist a closed simple curve $u:\mathbb S^1\to\mathbb R^...
- 60.5k
1
vote
1
answer
82
views
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 ...
- 439
2
votes
0
answers
70
views
Separating a certain planar region with an open set
I have a fairly specific question related to plane separation properties. I couldn't quite see how to use Phragmen–Brouwer properties to answer it because those kind of results generally apply to ...
- 517
8
votes
1
answer
518
views
Convergence in the Caratheodory sense and Hausdorff sense
Among Jordan domains, I understand that Caratheodory convergence is weaker than Hausdorff convergence.
But if a sequence of Jordan domains all have rectifiable boundary whose arc length are all $L$, ...
- 239
4
votes
1
answer
278
views
Nowhere compact subsets of the plane
Suppose $X\subseteq \mathbb R^2$ is nowhere compact ($X$ has no compact neighborhood) and non-empty.
Can $X$ be densely embedded into the plane?
In other words, is there a dense set $X'\subseteq ...
- 3,317
6
votes
2
answers
192
views
A non-Borel union of unit half-open squares
On the complex plane $\mathbb C$ consider the half-open square $$\square=\{z\in\mathbb C:0\le\Re(z)<1,\;0\le\Im(z)<1\}.$$
Observe that for every $z\in \mathbb C$ and $p\in\{0,1,2,3\}$ the set $...
- 41.8k
10
votes
1
answer
315
views
Fractal plane continuum with order $\aleph_0$?
Continuum means compact and connected.
Define the order of a point $x$ in a continuum $X$ to be the least cardinal $\alpha$ such that $X$ has a neighborhood base of open sets at $x$ with no more ...
- 3,317
1
vote
0
answers
142
views
Comparing Different Notions of Unicoherence in the Plane
Unicoherence is a generalization of simple connectedness that has been useful in topology in one and two dimensions. It is also a fundamental concept in shape theory, and thus has relations to Cech ...
- 439
5
votes
1
answer
186
views
A compact set in the plane with small sum-set and large projections
Problem. Let $K$ be a compact subset of the plane such that the projection of $K$ on each line has non-empty interior in the line. Has $K+K$ or $K-K$ non-empty interior in the plane?
Remark. The ...
- 41.8k
4
votes
1
answer
575
views
A question on Möbius strip and Jordan curve
If $A\subset \Bbb R^2$ then is the following statement true?
$\{(x,y)\in {(A\times A)/ \sim}\,\,\,|\,\, (x,y)\sim(y,x)\}\simeq$ Möbius strip $\iff A$ is a Jordan curve.
- 4,195
1
vote
1
answer
158
views
Trouble with plane embedding
Let $C$ be the middle-thirds Cantor set. Obviously $C\times [0,1]$ embeds into the plane. But $C\times D$ does not, $D$ being a closed disc in the plane.
Are there any general results which can be ...
- 3,317
17
votes
1
answer
988
views
Can two-point sets be Borel?
Recall that a two-point set is a subset of the plane which meets every line in exactly two points. Such a set was first constructed by Mazurkiewicz in 1914.
I wonder if the following question of ...
- 32.1k
13
votes
5
answers
1k
views
Connectedness in the plane
There are several open problems in topology which concern connectedness and subsets of the plane. The biggest of these is undoubtedly:
Question. Does every non-separating plane continuum have the ...
- 955
27
votes
3
answers
3k
views
A question about subsets of plane
Is there a subset $X$ of plane with two points $x, y$ such that each one of $X \setminus \{x\}$, $X \setminus \{y\}$ is isometric to $X$? I tried hard to construct a counterexample but failed.
Sorry ...
- 271
4
votes
0
answers
352
views
A generalized ellipse
We know that an ellipse is the locus of all point $z$ in the plane with $$|z-a|+|z-b|=\lambda$$
where $a,b$ are two given points in the plane and $\lambda$ is a constant.
Now we consider the ...
- 356
30
votes
3
answers
2k
views
Is there a subset of the plane that meets every line in two open intervals?
Using the Axiom of Choice, it is possible to construct a subset of the plane that meets every line in two points (these are called "$2$-point sets"). What if, instead of points, we ask for two open ...
- 18.5k
12
votes
1
answer
861
views
Connected components $0-1$ matrices
Let $M$ be a $0-1$ matrix.
Here a matrix has one component means we can traverse from a matrix entry $(i,j)$ which is $1$ to any other one by moving step of $(i\pm1,j),(i,j\pm1),(i\pm1,j\pm1)$ where ...
- 13.9k
10
votes
1
answer
346
views
A forked plane continuum
I came up with this question while trying to solve the following MO one:
Does every connected set that is not a line segment cross some dyadic square?
Suppose $C$ is a plane continuum (i.e. a ...
- 1,375