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.

Filter by
Sorted by
Tagged with
12 votes
1 answer
91 views

Can we describe the connected components in the graph of $\cos\left(x\right)-\sin\left(x+y\right)=\cos\left(x^{2}y\right)$?

The graph of $\cos\left(x\right)-\sin\left(x+y\right)=\cos\left(x^{2}y\right)$ is, perhaps unsurprisingly, pretty wild. Here is a Desmos version of it; a few screenshots are below, showing different ...
user avatar
  • 21.5k
1 vote
1 answer
43 views

Is there a nontrivial LOTS that is connected and totally path disconnected?

By nontrivial LOTS I mean a linearly ordered space that contains more then one point. Being totally path disconnected means that every path in the space is constant. A connected linearly ordered ...
user avatar
1 vote
0 answers
27 views

Proof verification: Demonstrating that the unit circle is path-connected

I'm trying to show that the unit circle is path-connected. I have a sketch of a proof, and was hoping someone could tell me if this is on the right track. Given points $(a,b), (m,n)$ on the circle, I ...
user avatar
1 vote
0 answers
23 views

Connected complement of compact subset

Let $U$ be an open subset of $\mathbb{R}^n$ and $K$ a compact subset of $U$ such that $U\setminus K$ is connected. Does there exist an open set $V$ such that $K\subseteq V \subseteq \overline{V} \...
user avatar
0 votes
0 answers
28 views

Showing that K-topology is not path-connected without using compactness

Let $\mathbb{R}_K$ denote the real line in the K-topology, which is the topology generated by the basis $\left\{(a,b)|a,b \in \mathbb{R}\right\} \cup \left\{(a,b)-K|a,b \in \mathbb{R}\right\}$, where $...
user avatar
0 votes
2 answers
36 views

Artin, Chapter 2, Misc.6

I am trying to solve miscellaneous exercise 6 in Chapter 2 of Artin's book, Algebra. Below is the statement of the problem. Let $a = (a_1, \ldots, a_k)$ and $b = (b_1, \ldots, b_k)$ be points in $k$-...
user avatar
1 vote
1 answer
47 views

Show that $\{(x,y): 0 < x \leq 1; y = \sin{1/x}\} \cup \{z: x = 0; -1 \leq y \leq 1\}$ is not path connected [duplicate]

Show that $\{(x,y): 0 < x \leq 1; y = \sin{1/x}\} \cup \{z: x = 0; -1 \leq y \leq 1\}$ is not path connected. The book gives this set as an example of a closed, connected but not path connected set....
user avatar
  • 493
-1 votes
1 answer
34 views

Question on the connectedness of the orthogonal group

I want to show that the quotient $O_2^- = O_2/SO_2$ is connected. My idea was as follows: It's easy to show that $SO_2$ is connected. $S0_2$ is a topological group (normal subgroup of a topological ...
user avatar
  • 337
0 votes
1 answer
40 views

Can a simply-connected topological space be written as the union of the images of all possible loops at a given basepoint?

Can a simply-connected topological space be written as the union of the images of all possible loops at a given basepoint? Let $X$ be a simply-connected topological space, fix $x\in X$, and let $\{\...
user avatar
1 vote
1 answer
53 views

Prove or disprove that a Jordan curve exists that contains two points.

Question (not homework): Let $\Omega$ be open and connected. Prove or disprove that if $z_1, z_2\in\Omega$ are distinct, then there exists a Jordan curve $\gamma\subseteq\Omega$ such that $z_1, z_2\...
user avatar
2 votes
1 answer
43 views

Is X connected by paths?

I have the following space $X=[0,1) \cup \{y,z\}$ with $y \neq z$ and $y,z \notin [0,1)$. Where the basic open for point in $[0,1)$ are taken as the basics to the topology that R inherits to the space....
user avatar
  • 97
1 vote
0 answers
75 views

How to show that $SO_{2,1}^+$ is path connected?

I need to show that $SO_{2,1}^+$ is path connected. First of all here are a couple of definitions we are using in the lecture: Let $n=r+s$ for $r,s \in \mathbb{N}$. $$I_{r,s} := diag(\underbrace{ 1,\...
user avatar
  • 2,464
0 votes
1 answer
59 views

Proving a topological space is path connected

Suppose $(X,T_X)$ is a topological space and $\infty_X \notin X$. Write $X^* = X \cup \{\infty_X\}$ and suppose the open sets of $X^*$ the empty set and the union of an open set in $X$ and the point $\...
user avatar
1 vote
0 answers
55 views

Exercise 8, Section 24 of Munkres’ Topology

(a) Is a product of path connected spaces necessarily path connected? (c) If $f:X \to Y$ is continuous and $X$ is path connected, is $f(X)$ necessarily path connected? (d) If $\{A_\alpha \}$ is a ...
user avatar
  • 1,381
2 votes
1 answer
30 views

Non simply-connected covering space of two other non simply-connected covering spaces

Let $X$ be a space with fundamental group $\mathbb{Z}$ which is path-connected, locally path-connected, and semilocally simply-connected, and let $x\in X$. We have two covering spaces $p_1: (\tilde{X}...
user avatar
1 vote
1 answer
52 views

Showing that equivalence class of path connected points is closed

Let $X\subseteq \mathbb{R}^{n} $ be open and $$[x]_{\sim} = \{y \in X \mid \text{there exists a continuous path from }x \text{ to }y \text{ in X}\}.$$ I want to show that $[x]_{\sim}$ is closed in/...
user avatar
  • 192
0 votes
1 answer
48 views

Example 6, Section 24 of Munkres’ Topology

The ordered square $I_0^2$ is connected but not path connected. My attempt: It’s easy to check $x\times (0,1)=(x\times 0, x\times 1)$, where $x\in I=[0,1]$. Let $f:[a,b]\to I_0^2 $ be a path from $0\...
user avatar
  • 1,381
1 vote
0 answers
61 views

Definition of Path Connected in Munkres’ Topology

Given point $x$ and $y$ of the space $X$, a path in $X$ from $x$ to $y$ is a continuous map $f:[a,b]\to X$ of one closed interval in the real line into $X$, such that $f(x)=a$ and $f(y)=b$. A space $X$...
user avatar
  • 1,381
1 vote
0 answers
22 views

Need help with a proof. Graph generation in an environment with obstacles.

Given is a set of points (PFeature) in a map which are at maximum distance from their closest obstacles. these points are obtained after generating the generalized voronoi diagram and applying a ...
user avatar
0 votes
1 answer
79 views

Example 4 & 5, Section 24 of Munkres’ Topology

Define the unit sphere $S^{n-1}$ in $\Bbb{R}^n$ by the equation $S^{n-1}=\{ x | \| x\|=1\}$. If $n\gt 1$, it is path connected. To be honest I don’t understand the proof, even after assuming $g$ is ...
user avatar
  • 1,381
1 vote
1 answer
64 views

If $X$ $\times$ $Y$ is path connected so is $X$ $\times$ $Y$ - $(x_0,y_0)$ for some $(x_0,y_0)$

I want to prove the following Let $X$ $\times$ $Y$ be path-connected show that $X$ $\times$ $Y$ $\setminus$ $(x_0,y_0)$ is still path connected for some point $(x_0,y_0)$ My attempt was by ...
user avatar
  • 600
1 vote
1 answer
31 views

Example for a set that is open connected but not path connected in an arbitrary topological space

I know the result that in $R^n$ any open connected set is path connected. But I guess in general the result is not true in an arbitrary topological space. Can anyone give an example showing this? ...
user avatar
  • 572
0 votes
1 answer
35 views

$(X,\tau_X)$ is a path connected, then is there a continuous function from $(X,\tau_X)\rightarrow(\mathbb{R},\tau_E)$ with , $f(x)=0$ and $f(y)=1$

I think the answer to this question is yes there exists. My main reasoning is that since there is a continuous path $\gamma: ([0,1],\tau_E)\rightarrow (X,\tau_X)$, with $\gamma(0)=x$ and $\gamma(1)=y$ ...
user avatar
  • 21
0 votes
1 answer
28 views

Path-connectedness swapped definition

Suppose $( X , τ_X )$ is path-connected with two distinct points $x \neq y$ . Then is it true that there is a continuous function from $( X , τ_X )$ to $([0,1],τ_𝐸)$ with $𝑓(𝑥)=0$ and $𝑓(𝑦)=1$ ?...
user avatar
  • 996
0 votes
3 answers
50 views

$C = (\{0\} \times [0,1]) \: \cup \: \{ (\frac{1}{n},y), n \in \mathbb{N}, y \in [0,1] \} \:\cup \: ([0,1] \times \{0\})$ is connected?

I want to prove wich $C = (\{0\} \times [0,1]) \: \cup \: \{ (\frac{1}{n},y), n \in \mathbb{N}, y \in [0,1] \} \:\cup \: ([0,1] \times \{0\})$ is connected. My attempt: $C$ is the union of sets. If we ...
user avatar
1 vote
0 answers
54 views

show that in $\mathbb{R}^2$ with the cofinite topology, every open set is both connected and path connected.

I want to show that in $\mathbb{R}^2$ with the cofinite topology, every open set is both connected and path connected. To do so, I thought of using the following result : for a finite set S, the set $...
user avatar
  • 171
1 vote
2 answers
79 views

Connectedness of Hartogs' Triangle in $\mathbb C^2$

Show that arbitrary $(z_1,z_2)$ in the Hartogs' triangle $$\triangle_H = \{(z_1,z_2) \in \Bbb C^2: |z_1| < |z_2| < 1\}$$ can be joined by a path (in $\triangle_H$) to $w = (0,r)$ for some fixed ...
user avatar
1 vote
1 answer
51 views

Trouble showing a union of sets is connected.

Exercise. Consider the open ball $S = \{(x,y) \in \Bbb R^2: (x-1)^2 + y^2 < 1\}$ and the singleton $\{(0,0)\}$. Show that $S \cup \{(0,0)\}$ is connected. My attempt. I know for a fact that $S$ is ...
user avatar
  • 599
1 vote
1 answer
70 views

Is $\{ (x, y) \in \mathbb{C}^2 : x^2 + y^2 = 1\}$ connected in $\mathbb{C}^2$?

Is $\{ (x, y) \in \mathbb{C}^2 : x^2 + y^2 = 1\}$ connected in $\mathbb{C}^2$ ? How do I approach this problem ? I think that that the fact that continuous functions map connected sets to connected ...
user avatar
1 vote
1 answer
72 views

Is this graph connected or path-connected?

I was thinking about a map f such that $Graph(f)$ is connected but it is not path connected, this is the map: $$f(x)=\left\{\begin{array}{cl} \sqrt{x^2-1} &\ x\in \mathbb{Q} \\ -\sqrt{x^2-1} &\...
user avatar
0 votes
1 answer
41 views

Understanding the definition of path connected subspace

If $X $ is a topological space and $ Y $ is a subspace of $ X$, then what does it mean when we say $Y$ is a path connected subspace of $X$? Does it mean that any two points $ x $ and $ y $ in $ Y $ ...
user avatar
  • 572
2 votes
2 answers
57 views

Locally connected for sub space topology of $\Bbb R^2$

Consider the following subspace $X:=\Bbb R^2\setminus(\Bbb Q\times \Bbb Q)$ of $\Bbb R^2$, where $\Bbb R^2$ with the usual topology. I would like to check this space in the terms of various kinds of ...
user avatar
  • 820
2 votes
1 answer
126 views

How to show that this space is not path connected?

I apologize if this question has been already asked, but I can't find this space anywhere. $$X = \left\{\left(0.5, 1\right] \times \{0\} \; \cup \; \bigcup_{n \in \mathbb{N}} \left\{ \left(x, \frac{x}{...
user avatar
  • 1,288
0 votes
2 answers
34 views

Path connected product topology implies every topology is path connected

Suppose that $(X_i, \tau_{X_i})$ are path-connected topological spaces for all $i \in I$. I know that the product $\Pi_{i \in I}X_i$ with its product topology is path-connected. But is the converse ...
user avatar
  • 996
2 votes
1 answer
36 views

Is connected component of sublevel set of continuous function always path connected?

Let $f:\mathbb{R}^n\mapsto\mathbb{R}$ be a Lipschitz continuous function. Let $S_a\triangleq \{x\in\mathbb{R}^n\,|\,f(x)\leq a\}$ be a sublevel set of $f$. If $S_a$ is connected, is it always path-...
user avatar
  • 626
0 votes
1 answer
44 views

Countable set with singleton is closed is not a pathwise connected

Let $X$ be a countable set with topology $T$ such that the singleton is closed. Recall the following well-known fact. I think it was done by Sierpi'nski. Fact. $[0,1]$ can not be written as the union ...
user avatar
  • 2,153
0 votes
1 answer
38 views

Connected and disconnected space [duplicate]

$GL(2 , R) = \{ M | 2 × 2 \& \det M \neq 0\}$ be defined as subset of $\mathbb{R}^4$ For example Consider the matrix GL(2, R) = \begin{matrix} 1 & 2 \\ 3 & 4 \\ \end{matrix} then (1,2,...
user avatar
7 votes
2 answers
144 views

Suppose that $X$ is Hausdorff. Show that $X$ is locally path connected.

Let $f:[0,1]\rightarrow X$ be a continuous surjective function to a Hausdorff space $X$. Prove that $X$ has the following property: For every $x\in X$ and every neighborhood $U$ of $x$, there exists a ...
user avatar
  • 328
0 votes
2 answers
99 views

Prove that $X_1\cup X_2$ is path connected if and only if both $X_1$ and $X_2$ are path connected

Suppose $X_1\subset [0,\infty)\times\mathbb{R}$ and $X_2\subset (-\infty,0]\times\mathbb{R}$ such that $X_1\cap X_2=\{0\}\times[0,1]$. Prove that $X_1\cup X_2$ is path connected if and only if both $...
user avatar
  • 328
0 votes
0 answers
48 views

$W\setminus B_1$ is path-connected

I am reading the article $\textit{A fake topological Hilbert space}$ and I am trying to understand the proof of the following lemma: Lemma 3.6 Let $B_1$ and $B_2$ be $\sigma$-$Z$-sets in $Q=\prod_{1}^...
user avatar
  • 688
1 vote
1 answer
37 views

Does it matter where we start on an Euler graph?

For a graph to be an euler graph, all the vertices have to be of even degree...I was wondering "does which vertex to start matter?"...I played around and now I kind of think it doesn't ...
user avatar
  • 57
1 vote
1 answer
25 views

Show that $M = \{(x,y) \in \mathbb{R}^p \times \mathbb{R}^q : \lvert x \rvert = \lvert y \rvert \neq 0 \}$ is connected for $p,q \geq 2$

How can I show that $M = \{(x,y) \in \mathbb{R}^p \times \mathbb{R}^q : \lvert x \rvert = \lvert y \rvert \neq 0 \}$ is connected? I can find a path $\gamma(t) = (tx, ty)$ which lays on $M$ for $t$ ...
user avatar
  • 337
1 vote
1 answer
33 views

Shortest path with jumps (dynamic Bayesian network)?

Suppose I have the following graph structure: It has the following properties: There are four states $\mathcal{S} = {q,s_1,s_2,s_3}$ where $q$ is some origin state where we start from (though it is ...
user avatar
  • 632
0 votes
1 answer
24 views

Study some topological properties of $I^{\aleph_0}\times I^2/M$

I've been solving some problems from my Topology course, and I'm unable to finish this one: Given $I^{\aleph_0}$ the Hilbert cube, and given $Y=I^2/M$ the quotient space where $M=\{(0,0),(1,0),(0,1),(...
user avatar
  • 3,430
0 votes
2 answers
94 views

Show that $A$ is not path-connected.

I have the following problem: In $\mathbb{R}^2$ with the usual topology consider the set $A= \{(x,y)\in\mathbb{R}^2:x\in\mathbb{R}\setminus\mathbb{Q},y\geq 0\}\cup\{(x,y)\in\mathbb{R}^2:x\in\mathbb{Q},...
user avatar
1 vote
1 answer
44 views

Are function spaces between CW complexes delta-generated?

Let $X$ and $Y$ be CW complexes. I wish to consider the mapping space $\mathbf{map}(X,Y)$ of continuous maps $X\to Y$, equipped with the (compactly generated version of the) compact-open topology. I ...
user avatar
  • 1,944
2 votes
0 answers
35 views

Unsure about justification : First fundamental form and rectangles

I am trying to prove the following equivalence for a parametrized surface $f: U \to \mathbb{R}^3 $ with $U = (0,A) \times (0,B) $ : For every Rectangle $R$ in $U$, the opposite sides of $f(R)$ have ...
user avatar
  • 336
1 vote
1 answer
64 views

Is $S=\{(x,y,z)\in\Bbb R^3\mid x^2+y^2\ge z\}$ path-connected?

Is $S=\{(x,y,z)\in\Bbb R^3\mid x^2+y^2\ge z\}$ path-connected? My thoughts: I think it is, but I can't construct a path from arbitrary points $p,q\in S$ explicitly. If we imagine the curve $f(x,y)=x^...
user avatar
0 votes
1 answer
50 views

Is $(\mathbb{R}^2, d)$ connected?

$\mathbb{R}^2$ with the metric: $$d:\mathbb{R}^2\times \mathbb{R}^2\to \mathbb{R}$$ defined by $d(x,y)=0,$ if $(x_1,y_1)=(x_2,y_2)$ and $d(x,y)=|x_1|+|x_2|+|y_1-y_2|$ if $(x_1,y_1)\neq(x_2,y_2),$ they ...
user avatar
0 votes
0 answers
54 views

Fundamental Group of Polygons

In Kosniowski's book "A First Course in Algebraic Topology", he used Van Kampen's theorem to calculate the fundamental group of given pictures. He defined the first open set, for example for ...
user avatar

1
2 3 4 5
12