# 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.

63
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### 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 ...

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0
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163
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### 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 ...

6
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1
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237
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### 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?

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1
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55
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### 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
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1
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196
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### 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 ...

2
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2
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61
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### 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 \...

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0
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63
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### 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 ...

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2
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150
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### 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 ...

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2
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447
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### 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 $P$. ...

4
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### $1$-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 ...

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133
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### 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 ...

6
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1
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299
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### 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 ...

2
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1
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292
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### 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 ...

0
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1
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100
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### 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 ...

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### 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 ...

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### 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
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454
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### 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 ...

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0
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65
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### 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 ...

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0
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147
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### 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]\...

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### 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 ...

13
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1
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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. ...

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### 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 ...

2
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0
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### 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 ...

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1
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118
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### 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

1
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### 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 ...

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850
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### 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:...

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0
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### 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 ...

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1
answer

534
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### 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
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0
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154
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### 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
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### 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; \...

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### 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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0
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106
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### 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 ...

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2
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266
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### 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
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1
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275
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### 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 ...

2
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1
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678
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### 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 ...

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0
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### 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
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1
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### 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 ...

2
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1
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### 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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0
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### 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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2
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331
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### 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
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1
answer

329
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### 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....

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858
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### 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 ...

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### 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 ...

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1
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242
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### 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 ...

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2
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626
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### 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)...

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0
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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 $d(x, x_1)$, ...

4
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2
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### 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 ...

20
votes

1
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### Connected and locally connected, but not path-connected

Allow me to use some non-standard terminology:
A h-contractible space is a non-empty topological space $X$ such that, for any topological space $T$ and any pair of continuous maps $f_0, f_1 : T \to X$...

3
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632
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### 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 (...

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218
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### 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.
...