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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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?
20 questions's user avatar
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20 votes
1 answer
2k views

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 $f_0, f_1 : T \to X$...
Zhen Lin's user avatar
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18 votes
5 answers
91k 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 ...
JesseStimpson's user avatar
13 votes
2 answers
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 ...
Vaughn Climenhaga's user avatar
13 votes
1 answer
763 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. ...
Jeremy Brazas's user avatar
12 votes
1 answer
845 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 ...
Turbo's user avatar
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12 votes
2 answers
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 $\...
Portland's user avatar
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11 votes
2 answers
58k 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 ...
Goody Two Shoes's user avatar
10 votes
1 answer
2k views

Is a space with no covering spaces simply connected?

Suppose $X$ is a path connected space such that every connected covering space of $X$ is trivial (1-fold.) Must $X$ be simply connected? Intuitively, the answer seems to be no (imagine taking a disk,...
David Cohen's user avatar
10 votes
1 answer
508 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 ...
Taras Banakh's user avatar
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9 votes
1 answer
773 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 ...
Douglas Sirk's user avatar
8 votes
2 answers
493 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 ...
Jeremy Brazas's user avatar
8 votes
0 answers
691 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 ...
user16557's user avatar
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6 votes
1 answer
295 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 ...
Jeremy Brazas's user avatar
6 votes
1 answer
180 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's user avatar
6 votes
2 answers
439 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 ...
rvdaele's user avatar
  • 71
5 votes
2 answers
3k 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 ...
skupers's user avatar
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5 votes
2 answers
1k 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 ...
Simon's user avatar
  • 53
5 votes
2 answers
617 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)...
Portland's user avatar
  • 2,752
5 votes
2 answers
410 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 $P$. ...
PMCosmin's user avatar
5 votes
1 answer
313 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; \...
user1101010's user avatar
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 ...
user avatar
4 votes
1 answer
128 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 ...
erz's user avatar
  • 5,385
4 votes
1 answer
201 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 ...
erz's user avatar
  • 5,385
4 votes
1 answer
2k 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 ...
Jennifer Gao's user avatar
4 votes
0 answers
88 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 ...
Eduardo Longa's user avatar
4 votes
0 answers
344 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?
Biller Alberto's user avatar
3 votes
1 answer
194 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 ...
user1101010's user avatar
3 votes
1 answer
189 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 ...
D.S. Lipham's user avatar
  • 3,055
3 votes
1 answer
589 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 (...
user21816's user avatar
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3 votes
0 answers
105 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 ...
PtH's user avatar
  • 280
2 votes
2 answers
327 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 ...
Transcendental's user avatar
2 votes
2 answers
61 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 \...
Dominic van der Zypen's user avatar
2 votes
1 answer
933 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 ...
Jason Rute's user avatar
  • 6,237
2 votes
2 answers
3k 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 ...
hurtledown's user avatar
2 votes
1 answer
611 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 ...
Alex M.'s user avatar
  • 5,207
2 votes
2 answers
147 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 ...
D.S. Lipham's user avatar
  • 3,055
2 votes
1 answer
279 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 ...
Zixun Tau's user avatar
2 votes
1 answer
175 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 $f$. $f$ ...
Jürg W. Spaak's user avatar
2 votes
1 answer
239 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 ...
thogrhm's user avatar
  • 23
2 votes
0 answers
58 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 ...
Leon Staresinic's user avatar
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 ...
J.K.T.'s user avatar
  • 487
2 votes
0 answers
140 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 ...
Pietro Majer's user avatar
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2 votes
0 answers
61 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 ...
user1101010's user avatar
2 votes
0 answers
152 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. ...
user1101010's user avatar
2 votes
0 answers
116 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 ...
QuantumLogarithm's user avatar
1 vote
1 answer
115 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 ...
user1101010's user avatar
1 vote
1 answer
326 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....
Ali Taghavi's user avatar
1 vote
0 answers
63 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 ...
mathcounterexamples.net's user avatar
1 vote
0 answers
115 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 ...
Portland's user avatar
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