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.
564
questions
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 ...
- 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 ...
- 709
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 ...
- 297
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} \...
- 173
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 $...
- 228
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 $k$-...
- 297
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....
- 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 ...
- 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 $\{\...
- 1,072
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\...
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....
- 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 $r,s \in \mathbb{N}$.
$$I_{r,s} := diag(\underbrace{ 1,\...
- 2,464
0
votes
1
answer
59
views
Proving a topological space is path connected
Suppose $(X,T_X)$ is a topological space and $\infty_X \notin X$. Write $X^* = X \cup \{\infty_X\}$ and suppose the open sets of $X^*$ the empty set and the union of an open set in $X$ and the point $\...
- 586
1
vote
0
answers
55
views
Exercise 8, Section 24 of Munkres’ Topology
(a) Is a product of path connected spaces necessarily path connected?
(c) If $f:X \to Y$ is continuous and $X$ is path connected, is $f(X)$ necessarily path connected?
(d) If $\{A_\alpha \}$ is a ...
- 1,381
2
votes
1
answer
30
views
Non simply-connected covering space of two other non simply-connected covering spaces
Let $X$ be a space with fundamental group $\mathbb{Z}$ which is path-connected, locally path-connected, and semilocally simply-connected, and let $x\in X$.
We have two covering spaces $p_1: (\tilde{X}...
- 1,072
1
vote
1
answer
52
views
Showing that equivalence class of path connected points is closed
Let $X\subseteq \mathbb{R}^{n} $ be open and $$[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/...
- 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\...
- 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$...
- 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 ...
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 ...
- 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 ...
- 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?
...
- 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$ ...
- 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$ ?...
- 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 ...
- 33
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 $...
- 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\}$$ can be joined by a path (in $\triangle_H$) to $w = (0,r)$ for some fixed ...
- 10.3k
1
vote
1
answer
51
views
Trouble showing a union of sets is connected.
Exercise. Consider the open ball $S = \{(x,y) \in \Bbb R^2: (x-1)^2 + y^2 < 1\}$ and the singleton $\{(0,0)\}$. Show that $S \cup \{(0,0)\}$ is connected.
My attempt. I know for a fact that $S$ is ...
- 599
1
vote
1
answer
70
views
Is $\{ (x, y) \in \mathbb{C}^2 : x^2 + y^2 = 1\}$ connected in $\mathbb{C}^2$?
Is $\{ (x, y) \in \mathbb{C}^2 : x^2 + y^2 = 1\}$ connected in $\mathbb{C}^2$ ?
How do I approach this problem ?
I think that that the fact that continuous functions map connected sets to connected ...
1
vote
1
answer
72
views
Is this graph connected or path-connected?
I was thinking about a map f such that $Graph(f)$ is connected but it is not path connected, this is the map:
$$f(x)=\left\{\begin{array}{cl} \sqrt{x^2-1} &\ x\in \mathbb{Q} \\
-\sqrt{x^2-1} &\...
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 $ ...
- 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 ...
- 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}{...
- 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 ...
- 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-...
- 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 ...
- 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,...
- 17
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 ...
- 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 $...
- 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}^...
- 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 ...
- 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$ ...
- 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 ...
- 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),(...
- 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},...
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 ...
- 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 ...
- 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^...
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 ...
- 842
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 ...
- 395