# 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.

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### 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. ...
621 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 ...
52 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 ...
67 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 ...
182 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 ...
313 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:...
103 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 ...
352 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 ...
140 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. ...
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### 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 $d(x, x_1)$, ...
613 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 ...
340 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 (...
212 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. ...
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### 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 ...
365 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
831 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 ...
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### 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 ...
43k 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 ...
### 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?