Questions tagged [path-connected]
For questions relating to path-connected topological spaces, that is, spaces where any two points can be connected by a path.
66 questions
0 votes
0 answers
45 views
Name for "degree" of strong connectivity
A pair of vertices $a$ and $b$ are in the same strongly connected component of a digraph iff there is path from $a$ to $b$ and also a path from $b$ to $a$ in that component. Question: is there an ...
- 14k
8 votes
1 answer
194 views
About path-connected components of the Bohr compactification of $\mathbb{R}^d$
Let ${\rm b}(\mathbb{R}^d)$ denote the Bohr compactification of $\mathbb{R}^d$, with $d\in\mathbb{N}$. This is the Pontryagin dual of the group $\mathbb{R}^d_d$, corresponding to $\mathbb{R}^d$ with ...
- 323
0 votes
1 answer
185 views
Product of adjacency matrices and connected graphs
Let $G$ and $ H$ be two graphs on the same vertex set (i.e. $V(G)=V(H)$). Suppose that $K$ is the graph with adjacency matrix $A_G\times A_H$, where $A_G$ and $A_H$ are adjacency matrix of $G$ and $H$,...
- 247
2 votes
1 answer
237 views
A topological characterization of trees?
Motivated by this complex dynamics question: Let $X$ be a compact, path-connected metric space. Suppose there exist an integer $N\geq 2$ and distinct points $p_1,\dots,p_N\in X$ such that no proper ...
- 3,659
2 votes
0 answers
247 views
If $X$ is a strong deformation retract of $\mathbb{R}^n$, then is $X$ simply connected at infinity?
Let $X \subseteq \mathbb{R}^n$, and assume there is a strong deformation retract from $\mathbb{R}^n$ to $X$. Is $X$ necessarily simply connected at infinity? (Edit) Follow up question: if there is a ...
- 893
6 votes
1 answer
295 views
Planar compact connected set whose boundary has a finite length is arcwise connected
Let $K \subset \mathbb{R}^{2}$ be a compact connected set such that $\mathcal{H}^{1}(\partial K)<+\infty$. Is $K$ arcwise connected?
user524824
0 votes
1 answer
133 views
Let K be a compact set in a surface, U component of S-K, K'=S-U. K has finitely many components. Does every component of K' contains a component of K? [closed]
Let $S$ be a compact connected surface. Let $K$ be a compact subset of $S$ and suppose that $K$ has a finite number of connected components. Let $U$ be a connected component of $S \setminus K$ and ...
3 votes
1 answer
312 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 ...
- 3,691
2 votes
2 answers
107 views
Does $(\omega, E)$ with the cycle condition have an $\omega$-path?
Let $G = (V,E)$ be a simple, undirected graph. We say that $v\neq w\in V$ lie on a common cycle if there is an integer $n\geq 3$ and an injective graph homomorphism $f: C_n\to V$ such that $v,w\in \...
- 53.7k
2 votes
0 answers
93 views
Separating property of a finite union of topological disks
Let $X$ be a topological $2$-sphere. Let $D_1, D_2, \dots, D_n \subset X$ be a finite family of closed topological disks (i.e. sets homeomorphic to the closed unit disk). Let $\mathcal{U} = \bigcup_{1 ...
- 111
2 votes
2 answers
224 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 ...
- 3,691
5 votes
2 answers
566 views
Connectedness of Quot schemes
Let $X$ be a connected projective scheme over $\mathbb{C}$ and $E$ a coherent sheaf on $X$. Consider the Quot scheme $\operatorname{Quot}_X(E,P)$ of quotients of $E$ of fixed Hilbert polynomial $P$. ...
- 51
4 votes
0 answers
122 views
$1$-parameter family of minimal embeddings and the maximum principle
Let $M^3$ be a closed orientable smooth manifold and let $g_t$ be a (smooth) $1$-parameter family of Riemannian metrics on $M$, $t \in \mathbb{R}$. Let $P \subset M$ a fixed closed orientable embedded ...
- 2,127
4 votes
1 answer
196 views
Is every path connected $F_\sigma$ subset of a plane an image of $[0,1)$?
The title says it all. Let $A$ be a path connected $F_\sigma$ subset of a plane (or more generally $\mathbb{R}^n$). Recall that a subset is called $F_\sigma$ if it is a union of a sequence of closed ...
- 5,643
6 votes
1 answer
400 views
How complicated can the path component of a compact metric space be?
Let $X$ be a compact metric space and $P$ be a path component of $X$. Since we are not assuming $X$ is locally path connected, $P$ must need not be open nor closed. Certainly, $P$ must be separable ...
- 7,710
2 votes
1 answer
349 views
Opposite-nearest neighbor algorithm vs. nearest neighbor algorithm
Take the traveling salesman problem, but with three slight twists: You can choose a different start vertex for each of the two algorithms. Each path from one vertex to another is of unique, arbitrary ...
- 21
0 votes
1 answer
150 views
Connectedness of the set having a fixed distance from a closed set 2
This question is related to this one: Connectedness of the set having a fixed distance from a closed set. Suppose $F$ is a closed and connected set in $\mathbb{R}^n$ ($n>1$). Suppose the complement ...
- 411
2 votes
0 answers
87 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
4 votes
0 answers
419 views
When every closed and connected subset is path connected
Let $X$ be a compact $T_0$ topological space such that its closed and connected subsets are path connected. Is there any characterization for such a space?
6 votes
2 answers
563 views
Gromov Hausdorff distance to tubular neighborhood
Let $M$ be a compact path metric space in $\mathbb{R}^d$, and for $\sigma>0$, $$ M_\sigma:=\{y\in\mathbb{R}^d:\min_{x\in M}\|x-y\|\leq\sigma\} $$ the $\sigma$-tube around $X$ in $\mathbb{R}^d$. I ...
- 71
1 vote
0 answers
84 views
Connected components of bounded linear operators of $V = (\mathcal C(U(1), \mathbb C) , \lVert \cdot \rVert_\infty)$
This question is related to this one. Consider the complex Banach space $V=(\mathcal C(U(1), \mathbb C), \Vert \cdot \Vert_\infty)$ where $\mathcal C(U(1), \mathbb C)$ is the space of continuous ...
- 1,375
0 votes
0 answers
191 views
Derivative of matrix argument function with respect to eigenvalues of argument
Let $\mathsf{SPD}_n$ denote the set of all real symmetric and positive definite $n\times n$ matrices. This set is convex so for every $A,B\in\mathsf{SPD}_n$ there exists a smooth path $\varphi:[0,1]\...
- 105
2 votes
0 answers
191 views
Is a closed connected semilattice of $C(I)$ path-connected?
Let $\Gamma $ be a sub-lattice of the Banach space $\big( B(S),\|\cdot\|_\infty\big)$ of all bounded real valued functions on the set $S$ (meaning that for any $f,g\in\Gamma $ both functions $f\wedge ...
- 63.5k
13 votes
1 answer
876 views
Is there a compact, connected, totally path-disconnected topological group?
There exist homogeneous spaces such as the pseudo-arc, which are compact, connected, and totally path-disconnected. Is there a nontrivial, Hausdorff topological group with the same properties, i.e. ...
- 7,710
9 votes
1 answer
895 views
Lorenz attractor path-connected?
Can we tell if the Lorenz attractor is path-connected? By the attractor I do not mean only the line weaving around, but rather its closure. EDIT: The answer below is unsatisfactory, and possibly ...
- 321
2 votes
0 answers
80 views
Does there exist a continuous path between two sets of oriented basis for a vector space out of a collection of subspaces?
This is cross post to the question at MSE. Let $V_1, V_2, \dots, V_n$ be a collection of vector subspaces in $\mathbb R^n$. For each $j=1, \dots, n$, $\dim(V_j) = m$ with $2 \le m < n$. We also ...
- 195
1 vote
1 answer
155 views
Is there a homeomorphism between the sets of Schur stable and Hurwitz stable matrices in companion forms?
This is a cross-post to the question I asked at MSE. The set of Schur stable matrices is \begin{align*} \mathcal S = \{A \in M_n(\mathbb R): \rho(A) < 1\}, \end{align*} where $\rho(\cdot)$ denotes ...
- 195
3 votes
1 answer
234 views
Can we classify the general linear maps such that for a fixed matrix $A \in M_n(\mathbb R)$ the conjuagation action fixes the first column?
Let $A \in GL_n(\mathbb R)$ be fixed. Let us consider the conjugation action by $G \in GL_n(\mathbb R)$, i.e., $G^{-1}AG$. I would like to see a way to identify the matrices such that the action fixes ...
- 195
-1 votes
1 answer
926 views
Find all paths on undirected Graph [closed]
I have an undirected graph and i want to list all possible paths from a starting node. Each connection between 2 nodes is unique in a listed path is unique, for example give this graph representation:...
1 vote
0 answers
142 views
Path connected without bounded path connected subset?
Question: Is there a path connected subset of $\mathbb R^2$, without any bounded path connected subset (aside from singletons)? Motivation: If we replace "path connected" by "connected", then the ...
- 2,859
10 votes
1 answer
631 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 ...
- 44.3k
2 votes
0 answers
180 views
When does $S \cap GL_4(\mathbb R)$ have precisely two connected components where $S \subset M_4(\mathbb R)$ is a linear subspace?
This is a cross-post to the question I asked at MSE. Let $A \in M_4(\mathbb R)$ and $A = (e_2, x, e_4, y)$ where $e_2, e_4$ are standard basis in $\mathbb R^4$ and $x,y$ are undetermined variables. ...
- 195
5 votes
1 answer
344 views
Is the set $\{\zeta: \rho(A(\zeta))< 1\}$ connected for matrices under parameterization of first $m$ rows?
$\newcommand{\eqqcolon}{=\mathrel{\vcenter{:}}}$ Fix $n, m \in \mathbb N$ with $n > m$. Let $\zeta \in \mathcal{M}(m \times n; \mathbb C)$ and we fix a $\zeta_0 \in \mathcal M( (n-m) \times n; \...
- 195
2 votes
1 answer
220 views
Proof of existence and uniqueness of solution to f(c)=0
I have a function $f:R^n_+\rightarrow R^n$ for which I want to show the following: $$\exists c\in R^n_+ \quad \forall i,j\,\,f_i(c)=f_j(c)$$ Where $f_i (c)$ are the different coordinates of $f$. $f$ ...
- 139
3 votes
0 answers
122 views
Bound on change in relative length from 'well-behaved' Jacobian?
(This question was originally asked on Mathematics Stack Exchange, and sat there for several weeks with low views and no answers.) Let $\phi$ and $\gamma$ be rectifiable curves in the same length ...
- 300
0 votes
2 answers
336 views
Is the function $g$ always injective where $g$ is obtained by lipschitz re-parametrization
Suppose $(X,d)$ is a metric space with the nearest point property and $a,b \in X$ with $a \ne b$. Suppose there is a path of finite length in $X$ from $a$ to $b$ and let $m$ be the infimum of the ...
- 131
0 votes
1 answer
344 views
Does there always exists a path $g:[0,1] \rightarrow X$ from $f(0)$ to $f(1)$ that has the same image as $f$ and …?
Suppose $(X,d)$ is a metric space and $f:[0,1] \rightarrow X$ is a path in $X$ with non-zero finite length $L$. Then, does there always exist a path $g:[0,1] \rightarrow X$ from $f(0)$ to $f(1)$ that ...
- 131
2 votes
1 answer
841 views
Riemannian manifolds: every compact subset is contained in a connected relatively compact open subset [closed]
While working on some problem (not relevant here), it turned out to be convenient to be able to enclose arbitrary compact subsets in "nicer" compact subsets, hence the question: if $(M,g)$ is a ...
- 5,497
1 vote
0 answers
271 views
Connectedness of symmetric subgroup of simply connected Lie group
Let $G$ be a connected real simple Lie group, and $\tau$ be an involutive automorphism of $G$. Then $\tau$ defines a symmetric subgroup $G^\tau$ of $G$. In general, $G^\tau$ is not necessarily ...
- 961
5 votes
1 answer
265 views
Inscribing a "chain" into an open cover
Let $X$ be a locally connected topological space, which is covered by open sets $\{U_{\alpha},\alpha\in A\}$ and let $C$ be an arc in $X$, i.e. a homeomorphic image of an interval. Is it always ...
- 5,643
2 votes
1 answer
1k views
Is every path connected space continuously path connected
Recall a topological space $X$ is path connected if for all $x,y \in X$ there is a continuous function $f\colon [0,1] \to X$ such that $f(0)=x$ and $f(1)=y$. Say that $X$ is continuously path ...
- 6,375
2 votes
0 answers
135 views
Number of self avoiding paths which are not ``tie together''
Consider the lattice $\mathbb{Z}^d$. Let $A_{n}$ be the set of returning self-avoiding paths (from $0$ to $0$) having length $n$. For any path $\omega \in A_{n}$, let $f(\omega)$ be the number of ...
- 295
2 votes
2 answers
390 views
A Jordan Separation Theorem for Polyhedral Surfaces
Let me begin by defining what a polyhedral surface is. A path-connected subset $ P $ of $ \mathbb{R}^{3} $ is called a polyhedral surface iff it is the union of a finite collection $ \mathcal{C} $ of ...
- 1,396
5 votes
3 answers
645 views
Two questions on path connected spaces
Is it true to say that a compact hausdorff space $X$ is path connected if and only if for every continuous function $f:X\to \mathbb{C}$, we have $f(X)\subset \mathbb{C}$ is path connected? For a ...
- 280
12 votes
1 answer
959 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 ...
- 1
2 votes
1 answer
284 views
How to infer missing nodes from a path?
I have a first data set which is a list of train stops with coordinates (lat, lon), but not the "links" between the nodes/stops (this could thought of as a null or empty graph). I have a second data ...
- 23
-1 votes
1 answer
305 views
Complements of images of complex analytic sets
It is known that the complement of an analytic set is connected. In general, the complement of a proper complex analytic set in a connected complex manifold is an arcwise connected dense open set. My ...
- 1,045
5 votes
2 answers
816 views
Beyond Cantor's Teepee
From Counterexamples in Topology by Steen and Seebach (2nd edition) example 129 page 145 we have an example of connected and totally path-disconnected space. It is defined as follow: Fix $p= (1/2,1/2)...
- 2,859
1 vote
0 answers
96 views
Analogue of a path-connected subspace in the context of point processes
Given a set of points $S$ in some metric space, a pair of points $x, x'$ will be termed $\epsilon$-connected if they are connected by a series of points $x_1, \ldots, x_m \in S$ such that $d(x, x_1)$, ...
- 111
4 votes
2 answers
1k views
topological group that is connected and locally connected but not path-connected
Is there a ($\mathrm{T}_0$) topological group that is connected and locally connected but is not path-connected? This is a cross-post from MSE, since my question there was posted over three weeks ago ...
user5810