# 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.

291 questions
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### 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 ...
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### 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 ...
0answers
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### 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 ...
1answer
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### 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 $f$ ...
1answer
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### 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 ...
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### 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 ...
0answers
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### 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 ...
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### 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 ...
2answers
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### 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 ...
1answer
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### 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 ...
2answers
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### 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'...
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### 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 ...
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### 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 ...
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1answer
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### 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 ...
1answer
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### 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 ...
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2answers
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### 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-...
2answers
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### 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-...
1answer
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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 $\mathcal E \cap \mathcal U ... 1answer 39 views ### Need prove a Graphs with n >= 5 with a Euler path and without Hamilton Path and Hamilton Cycle exist. I already proved a Euler Tour can't exist because all degrees would have to equal a number divisible by 2 but a Euler Path requires two odd degrees. I still have to prove a graph exists that contains ... 2answers 91 views ### Show that the unit sphere in$\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)$is pathwise(=arcwise) ... 1answer 43 views ### Is this topological transformation group locally path connected? A surface is an oriented connected sum of$g\geq 0$tori, with$b \geq 0$open disks removed, and$n \geq 0$punctures in its interior. Let Aut$^+(S,\partial S)$denote the group (under composition) ... 0answers 27 views ### Connectedness of$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 ...
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