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Questions tagged [path-connected]

Use this tag for question on path-connected spaces and related notions. These include locally path-connected spaces, arcwise connected spaces and so on. For the more general notion, use the (connectedness) tag.

2
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1answer
59 views

Path Connectedness of Simply Connected Space Minus a Point

Suppose that $X$ is a simply connected topological manifold of dimension at-least $2$. Fix a point $x \in X$ and define $\tilde{X}\triangleq X-\{x\}$. How can I prove that the $0^{th}$ signular ...
0
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0answers
21 views

Arc connectedness implies connected

I've seen that are one or two questions like this, but I'm not fully convinced that are right. Both pretty much say the same thing, so I leave the easier to understand: path connectedness implies ...
0
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0answers
36 views

Path-Connected Components [duplicate]

How many path-connected components does $O_{2}(\mathbb{R})$ have? My Attempt: I believe that the answer is 2, but I'm not sure how to work with the definition and describing the path-connected ...
2
votes
1answer
47 views

Proof of path connectedness of $S^{n-1}$

I need to proof that $S^{n-1}:=\{x\in\mathbb{R}^{n}\, :\, ||x||=1\}$, $n>1$ is path connected. So, for all $x,y\in S^{n-1}$, I need to show a function $f:[a,b]\rightarrow S^{n-1}$ such that $f$ ...
2
votes
1answer
35 views

Connected Graph With Minimum Degree

$G$ is a connected graph with $100$ vertices, where vertices have minimum degree $10$. Show G has a path with $21$ vertices. I know that for a graph with minimum degree $n$, there has to be a path of ...
0
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1answer
19 views

Locally path connected and connected imply path connected

Let $X$ be a connected locally path connected space. I want to show that it is also path connected. Following a suggestion in this answer, fix $a\in X$ and consider the set $$U_a = \{x\in X : \text{...
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0answers
40 views

Graph $G$ with Degree $\ge 2$ must contain a cycle

Show that a finite graph with all vertices with degree $\geq 2$ has a cycle that contains a vertex which is non-adjacent to any other vertices not contained in the cycle. I know how to prove that ...
3
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0answers
60 views

Non-existence of pushout in homotopy category

I want to show that $S^1_{(0)}\leftarrow *\to S^1_{(1)}$ has no pushout in the homotopy category without using Eilenberg–MacLane spaces. In a first step, I want to show that if there is such a ...
2
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2answers
112 views

Do i imagine the linear (straight line) homotopy in a correct way?

Today i learned about the linear homotopy which says that any two paths $f_0, f_1$ in $\mathbb{R}^n$ are homotopic via the homotopy $$ f_t(s) = (1-t)f_0(s) + tf_1(s)$$ Am i right in imagining the ...
1
vote
1answer
46 views

How to prove that the $n$-sphere $S^n$ is path connected for $n\geq 1$?

How would I prove that the $n$-sphere $S^n$ is path connected for $n\geq 1$? I have seen this question. I do not really understand this section: For b) The application $\varphi$ from $\mathbb{R}^n ...
3
votes
1answer
160 views

Representation of $\overline{\mathbb{Q}}$ in One Dimension

Introduction: Below gives an approach to realize the algebraic numbers $\overline{\mathbb{Q}}\subseteq\mathbb{C}$ within a $1$-dimensional compact connected abelian group (solenoid). Essentially a ...
0
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1answer
32 views

Prove that the connected circulant regular graphs of degree at least three contain all even cycles.

This is the question I am trying to solve, but while researching about circulant graph I came across Paley's graph of order 13. Now clearly when looking at this graph which is an example of circulant ...
6
votes
1answer
273 views

Path-continuity and the Axiom of Choice

We tell the following to our Calc III students (usually for $\mathbf{R}^2$, and never so formally): Let $A$ be an open subset of $\mathbf{R}^n$, $a\in A$, $f$ a real-valued function on $A$ and $\...
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0answers
26 views

Show that a connected graph contains a path between given vertices [duplicate]

Let $G$ be a connected graph with minimal degree $\delta(G) = k \geq 1$. Show that $G$ contains a path $(x_1,\dots,x_k)$ such that $G\setminus\{x_1,\dots,x_k\}$ is also connected. This problem seems ...
0
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0answers
12 views

probabilty in matrix

Given m,n dimensions of a 2D matrix; (i,j) initial co-ordinates; (x,y) final co-ordinates. What is the probability of being at (x,y) after at most k steps if we start from (i,j) initially? We can ...
1
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0answers
66 views

How to show that a set is connected

Can someone please explain to me the idea of connectedness and how they relate to these sets? I understand the definition of connectedness, but I am struggling to relate the definition to these ...
0
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2answers
65 views

Number of connected components Invariant

Why is the number of connected components invariant under homeomorphisms? I know that connectedness, as well as path connectedness, are properties conserved through homeomorphisms. But why is this ...
0
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1answer
29 views

Hamiltonian circuit

Prove that a graph that posses a Hamiltonian circuit must have no pendant vertices. To prove this, each vertex in a graph, that also has a hamiltonian circuit, much acquire at least two edges in ...
0
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2answers
55 views

Let $X$ be a subspace of $\Bbb R^2$ consisting of points whose coordinates are both irrational. Prove that $X$ is path-connected [closed]

I have to prove that if $X$ is a subspace of $\Bbb R^2$ consisting of points whose coordinates are both irrational then $X$ will be path-connected. I think that it isn't connected by arcs, but I don'...
0
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0answers
39 views

A problem on path connectedness of unit ball in R*R

Actually, I am having a problem over the fact that as $f:[0,1]\rightarrow\mathbb{R}^{2}$ , $f(t)= (1-t)x + ty$ is continuous, $x,y\in\mathbb{R}^{2}$, then $f([0,1])$ is connected . Then as unit ball ...
2
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0answers
11 views

Path / Graph problem with X nodes looking for Y paths with the most similar length.

I have the following graph / path problem: There is exactly 1 start node and 1 end node. There are also X (in this case 7) nodes, each connected to all other nodes and the start and end node with ...
2
votes
3answers
79 views

Path-connectedness of the complement of a finite number of points

Let $X=\mathbb{R}^3\setminus\{x_1,...,x_n\}$, the complement of a finite number of points. I want to show that $X$ is path-connected. I know how to do this intuitively. Let $y,z\in X$, then $t y+(1-t)...
1
vote
1answer
35 views

Suppose that the sets $A_{1},A_{2} \subset \mathbb{R}^n $ are connected and that they are not disjoint. Prove that $A_{1} \cup A_{2}$ is connected.

Suppose that the sets $A_{1},A_{2} \subset \mathbb{R}^n$ are connected and that they are not disjoint. Prove that $A_{1} \cup A_{2}$ is connected. The section including this question contains this ...
1
vote
2answers
51 views

proving that graph G is a path

I want to prove that the connected graph $G$ with $\delta(G) = 1$ and each vertex has a degree of $1$ or $2$, is a path. Can you tell whether my proof is correct? Since $G$ is connected, there is a ...
0
votes
2answers
50 views

Intersection of path-connected sets in $\mathbb{R}^{n}$

Let $A, B \subseteq \mathbb{R}^{n}$ be two path-connected sets. Is it true that $A \cap B$ is also path-connected? (A subset of $\mathbb{R}^{n}$ is path-connected if every pair of points in $A$ can ...
0
votes
1answer
22 views

Finding an open simply connected subset in a punctured open simply connected set

Let $X$ be an open , simply connected (path connected with trivial first homotopy group) subset of $\mathbb R^2$. Let $0\in X$. Is it true that for every $p,q\in X\setminus \{0\}$, there is an open ,...
5
votes
1answer
182 views

Is this subset of $M_4(\mathbb R)$ connected?

Let us consider an affine structure $\star$ of $M_4(\mathbb R)$ which has following form \begin{align*} \begin{pmatrix} 0 & * & 0 & * \\ 1 & * & 0 & * \\ 0 & * & 0 &...
0
votes
1answer
12 views

Help with Proof for Connect Open Space implying Finitely path connectedness

I would appreciate help with my (sketch of a) proof for the following problem: Question: Let $E\subset \mathbb{R^n}$ be an open connected open set. Show that for any $x,y\in E$ there is a finite ...
1
vote
1answer
54 views

Connected component of $\{(x_1,…,x_n) \; | \; x_1^2+…+{x_{n-1}}^2-x_n^2 \neq 0\}$

The question is in the title, actually for $n=2$, it's okay, for $n=3$, I thought that : $\{(x_1,...,x_n) \; | \; x_1^2+x_2^2-{x_{3}}^2 < 0 \}, \{(x_1,...,x_n) \; | \; x_1^2+x_2^2-{x_{3}}^2 > 0 ...
0
votes
1answer
34 views

Relation between components and path components of a topological space X.

Theorem: If $X$ is a topological space, each path component of $X$ lies in a component of $X$. If $X$ is locally path connected, then the components and the path components of $X$ are the same. Proof ...
1
vote
1answer
55 views

A path-connected graph is connected as a graph

This question regards the difference between being path-connected as a topological space, i.e. "Every two point $x,y \in X$ can be connected with a path" and being connected in a graph sense: "A graph ...
0
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0answers
145 views

Topologist Sine Curve, connected but not path connected

So I have to show that $X=U∪V$, $U=\{(0,y),y\in[-1,1]\}$, $V=\{(x,sin({1 \over x}),x>0\}$ is connected but not path connected My proof: Assume that $X $ is path connected and let $f:[0,1]\...
1
vote
1answer
100 views

Prove that $S^2 \vee S^2$ is path connected?

Prove that $S^2 \vee S^2$ is path connected? Let $a$ and $b$ be two points in a topological space $X$. A path in $X$ from $a$ to $b$ is a continuous map $f$ from [0,1] to $X$ s.t $f$(0) = $a$ and $f$(...
1
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0answers
12 views

What is the way to go from point 1 to 2 while remaining in the region defined by half torus?

Basically this is a trajectory planning problem. As shown in figure below, how can I ensure that in moving from a point (x,y,z) to another point within torus, I don't get out of that torus region? go ...
2
votes
1answer
73 views

Yamabe's theorem proof

*I'm trying to make the proof of Yamabe's Theorem that says that an arcwise connected subgroup of a Lie group G is a Lie subgroup of G. I found the proof in Goto's article (https://www.ams.org/...
2
votes
1answer
71 views

the fundamental group of $X$ is the symmetric group $S_3$, then whether it has a universal cover?

Question: Suppose that $X$ is a path-connected space with $\pi_1(X)=S_3$, which is the 3-symmetric group. I just wonder that whether $X$ has a universal cover. Try: Based on Hatcher, $X$ has a ...
3
votes
2answers
87 views

Connected but not path-connected

Let $r:[0,\infty)\rightarrow\mathbb{R}$ be given by $r(t)=1+e^{-t}$. Let $S\subset \mathbb{C}$ be the image of the "spiral" curve $f:[0,\infty)\rightarrow\mathbb{C}$ given by $f(t)=r(t)e^{it}$. Then $\...
2
votes
2answers
174 views

Path-conectedness of open balls implies path-connectedness

Let $M$ be a subset of a metric space $(E,d)$. I have just proved that $M$ path-connectedness implies $M$ connectedness. Now I need to show that if $M$ is connected and every open ball in $E$ is path-...
19
votes
2answers
652 views

Geometry of the set of coefficients such that monic polynomials have roots within unit disk

We let $\pi$ be the bijection between coefficients of the real monic polynomials to the real monic polynomials. Let $a\in \mathbb R^n$ be fixed vector. Then \begin{align*} \pi(a) = t^n + a_{n-1} t^{n-...
9
votes
1answer
145 views

How many connected components could the intersection of $\{A \in M_n(\mathbb R): \rho(A) < 1\}$ and an affine subspace in $M_n(\mathbb R)$ have?

Let $\mathcal E = \{A \in M_n(\mathbb R): \rho(A) < 1\}$ where $\rho(\cdot)$ is the spectral radius and $\mathcal U$ be an affine space in $M_n(\mathbb R)$. If we assume $\mathcal E \cap \mathcal U ...
1
vote
1answer
39 views

Need prove a Graphs with n >= 5 with a Euler path and without Hamilton Path and Hamilton Cycle exist.

I already proved a Euler Tour can't exist because all degrees would have to equal a number divisible by 2 but a Euler Path requires two odd degrees. I still have to prove a graph exists that contains ...
2
votes
2answers
91 views

Show that the unit sphere in $\mathbb{R}^3$ is pathwise(=arcwise) connected?

I know this question has been already asked but no one does not address the definition of pathwise(=arcwise) connectedness as the following: A set $C$ in metric space $(M,d)$ is pathwise(=arcwise) ...
2
votes
1answer
43 views

Is this topological transformation group locally path connected?

A surface is an oriented connected sum of $g\geq 0$ tori, with $b \geq 0$ open disks removed, and $n \geq 0$ punctures in its interior. Let Aut$^+(S,\partial S)$ denote the group (under composition) ...
0
votes
0answers
27 views

Connectedness of $GL_n(\mathbf{R}^{+})

Want to show any two matrices in $GL_n(\mathbf{R})^+$, i.e. $n\times n$ matrices with positive determinant can be connected by a path. Now It being a manifold path connectedness and connectedness are ...
11
votes
0answers
232 views

Does there exist a continuous path between two sets of oriented basis for a vector space out of a collection of subspaces?

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 have the condition: for any collection of $\...
4
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2answers
83 views

Open connected subsets of path connected spaces

Does every open and connected subset of path connected topological space has to be path connected? Statement should be false as there is a similar theorem but for Euclidean spaces, however I can't ...
3
votes
1answer
104 views

Is the quotient space path-connected?

Let $\mathbb C^n = \mathbb C \times \mathbb C \times \dots \times \mathbb C$ be the complex space. We define the quotient space $\mathbb C^n_{sym}$ by $\mathbb C^n / \sim$ where $x \sim y$ if there ...
1
vote
1answer
23 views

Does this subset of $GL_2(\mathbb R)$ under Vandermonde-like parametrization have precisely two path-connected components?

Let us fix $2$ distinct nonzero real numbers $\lambda_1 < \lambda_2\in \mathbb R$. Let $A \in M_2(\mathbb R)$ be a $2\times 2$ matrix given by \begin{align} \label{eq:q} \tag{$\star$} \begin{...
2
votes
1answer
55 views

Is this subset of $GL_5(\mathbb R)$ under Vandermonde-like parametrization of square matrices path-connected?

Let us fix $5$ distinct nonzero real numbers $\lambda_1, \dots, \lambda_5\in \mathbb R$. Let $A \in M_5(\mathbb R)$ be a $5\times 5$ matrix given by \begin{align} \label{eq:q} \tag{$\star$} \begin{...
1
vote
1answer
37 views

Local connectedness and path-connectedness of a square $I\times I$.

I am trying this problem. 4-10(From (J.Lee) Intoduction to Topological manifold) Let $S$ be the square $I\times I$ with the order topology generated by the dictionary order (see Problem 4-6). ...