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.
61
questions
20
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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
votes
1
answer
2k
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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 $f_0, f_1 : T \to X$...
18
votes
5
answers
91k
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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 ...
13
votes
2
answers
1k
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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 ...
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. ...
12
votes
1
answer
845
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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 ...
12
votes
2
answers
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(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 $\...
11
votes
2
answers
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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 ...
10
votes
1
answer
2k
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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,...
10
votes
1
answer
508
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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 ...
9
votes
1
answer
773
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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 ...
8
votes
2
answers
493
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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 ...
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 ...
6
votes
1
answer
295
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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 ...
6
votes
1
answer
180
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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?
6
votes
2
answers
439
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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 ...
5
votes
2
answers
3k
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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 ...
5
votes
2
answers
1k
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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 ...
5
votes
2
answers
617
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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)...
5
votes
2
answers
410
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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 $P$. ...
5
votes
1
answer
313
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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 $\zeta_0 \in \mathcal M( (n-m) \times n; \...
4
votes
2
answers
1k
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topological group that is connected and locally connected but not path-connected
Is there a ($\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 ...
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 ...
4
votes
1
answer
201
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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 ...
4
votes
1
answer
2k
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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 ...
4
votes
0
answers
88
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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 ...
4
votes
0
answers
344
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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?
3
votes
1
answer
194
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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 ...
3
votes
1
answer
189
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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 ...
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 (...
3
votes
0
answers
105
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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 ...
2
votes
2
answers
327
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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 ...
2
votes
2
answers
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 \...
2
votes
1
answer
933
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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
votes
2
answers
3k
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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 ...
2
votes
1
answer
611
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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 ...
2
votes
2
answers
147
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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 ...
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 ...
2
votes
1
answer
175
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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 $f$.
$f$ ...
2
votes
1
answer
239
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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 ...
2
votes
0
answers
58
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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 ...
2
votes
0
answers
70
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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 ...
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 ...
2
votes
0
answers
61
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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 ...
2
votes
0
answers
152
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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. ...
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 ...
1
vote
1
answer
115
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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 ...
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....
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 ...
1
vote
0
answers
115
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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 ...