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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.

10
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1answer
352 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.
2
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0answers
44 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 ...
1
vote
1answer
49 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 ...
3
votes
1answer
173 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 ...
0
votes
1answer
99 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
0answers
88 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 ...
9
votes
1answer
277 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 ...
2
votes
0answers
137 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. ...
5
votes
1answer
287 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; \...
2
votes
1answer
143 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 $f$. $f$ ...
3
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0answers
95 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 ...
0
votes
1answer
53 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 ...
0
votes
1answer
131 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 no-zero finite length $L$. Then, does there always exists a path $g:[0,1] \rightarrow X$ from $f(0)$ to $f(1)$ that ...
1
vote
1answer
109 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 ...
1
vote
0answers
120 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 ...
4
votes
1answer
143 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 ...
-1
votes
1answer
2k views

What exactly are Weakly connected graphs? [closed]

I see the definition for the Weakly connected graphs as : In a directed graph, the graph is weakly connected if there exists a path between any pair of nodes, without following the edge directions ...
2
votes
1answer
468 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 ...
2
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0answers
102 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 ...
2
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2answers
222 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 ...
18
votes
0answers
849 views

Unital $C^{*}$ algebras which all elements have path connected spectrum

A unital $C^{*}$ algebra is called "Path connected" if the spectrum of all its elements is a path connected subset of $\mathbb{C}$. Is the tensor product of two path connected algebra, a path ...
1
vote
1answer
248 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? 2....
12
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1answer
714 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 ...
2
votes
1answer
131 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 ...
-1
votes
1answer
155 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 ...
5
votes
2answers
434 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)...
1
vote
0answers
67 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 $d(x, x_1)$, ...
3
votes
1answer
423 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 ...
3
votes
1answer
276 views

When is a sublevel set path-connected?

I am trying to completely characterize the conditions on $f : \mathbb{R}^n \to \mathbb{R}$ under which $\{x | f(x) \le 0 \}$ is path-connected. There are many obvious conditions that are sufficient (...
0
votes
0answers
191 views

When is $\{ x \ge 0 | f(x) \le 0\}$ path-connected?

I'm trying to determine the conditions on $f : \mathbb{R}^n_{\ge 0} \to \mathbb{R^n}$ under which $\{ x \ge 0 | f(x) \le 0 \}$ is path-connected. We can assume that $f$ is continuous and concave. ...
2
votes
2answers
2k views

number of totally different path between two nodes in graph theory

I have an undirected, unweighted graph representing a network. I have a starting node and an end one. My 'network' is reliable if there is no node such that without that node s and t are not reachable ...
7
votes
2answers
344 views

Refining open covers in locally path connected spaces

Suppose $X$ is a locally path connected topological space and $\mathcal{U}$ is an open cover of $X$ (consisting of path connected sets if we want). One often wants the intersection $A\cap B$ of ...
3
votes
1answer
915 views

Connected level sets

This may be an ill-posed question, but suppose I have a collection of continuous, bounded, scalar-valued nonnegative functions $f_1(x,y),\dots,f_n(x.y)$ defined on the closed unit disk. Given a ...
8
votes
0answers
668 views

Path connected set of matrices?

Consider the collection of $n$ by $n$ matrices $$S=\{ A: A_{ij}\le0,\quad (-1)^{c_i}\det A(P_i;Q_i)<0 \quad \text{for} \quad i=1,\ldots, k\}$$ where $c_i\in \{0,1\}$, $P_i$ and $Q_i$ are disjoint ...
12
votes
2answers
1k views

Connectedness of space of ergodic measures

Let $X = \Sigma_p^+ = \{1,\dots,p\}^\mathbb{N}$ and let $f=\sigma\colon X\to X$ be the shift map. Let $\mathcal{M}$ be the space of Borel $f$-invariant probability measures on $X$ endowed with the ...
5
votes
2answers
732 views

Ends of topological spaces. Why independent of choice of ascending sequence of compact subsets?

Quoting from http://en.wikipedia.org/wiki/End_(topology): "Let X be a topological space, and suppose that K1 ⊂ K2 ⊂ K3 ⊂ · · · is an ascending sequence of compact ...
-3
votes
1answer
724 views

epsilon tube around continuous path in open set in R^n [closed]

This may have a simple answer, but I'm not getting anywhere. If $U$ is an open set in $\mathbb{R}^n$ (usual topology), and $p:[0,1] \to U$ is a continuous path, from $x=p(0)$ to $y=p(1)$, with $x,y \...
12
votes
2answers
2k views

(Path) connected set of matrices?

Let $N \in \mathfrak{M}_n(\mathbb{C})$ nilpotent, such that there exists $X \in \mathfrak M_n(\mathbb{C})$ with $X^2=N$ (take for instance $n>2$ and $N(1,n)=1$; $N(i,j)=0$ otherwise). Denote by $\...
15
votes
5answers
63k views

Finding all paths on undirected graph

I have an undirected, unweighted graph, and I'm trying to come up with an algorithm that, given 2 unique nodes on the graph, will find all paths connecting the two nodes, not including cycles. Here's ...
6
votes
2answers
36k views

Difference between connected vs strongly connected vs complete graphs [closed]

What is the difference between connected strongly-connected and complete? My understanding is: connected: you can get to every vertex from every other vertex. strongly connected: every vertex ...
4
votes
2answers
2k views

What does the property that path-connectedness implies arc-connectedness imply?

A space X is path-connected if any two points are the endpoints of a path, that is, the image of a map [0,1] \to X. A space is arc-connected if any two points are the endpoints of a path, that, the ...
18
votes
4answers
4k views

Can you explicitly write $\mathbb{R}^2$ as a disjoint union of two totally path disconnected sets?

An anonymous question from the 20-questions seminar: Can you explicitly write $\mathbb{R}^2$ as a disjoint union of two totally path disconnected sets?