All Questions
Tagged with path-connected metric-spaces
49 questions
4
votes
1
answer
62
views
Determine whether the set $A=\{(0,0)\neq (x,y)\in\mathbb R^2\mid x^2+y^2\leq |x|\}$ is connected.
This is an exercise from my Intro to Real Analysis class. The original questions asks to determine whether $A$ is closed, compact, and connected. I believe I was able to demonstrate closedness and ...
- 93
2
votes
0
answers
62
views
A $H^1$-$\sigma$-finite metric space that will have infinite $H^1$-measure under every homeomorphism
This is not a question, technically — apologies. It is an example that I really wanted to share.
Let $H^1$ denote the Haudorff measure of dimension 1.
Proposition: There exists a locally compact ...
- 5,379
0
votes
0
answers
19
views
Connected Metric Spaces: Strategies
I am not really sure if my ideas in this topic are correct. Can anyone help me?
Finding the connected components of a metric space $X$.
Suppose there are two connected components $C_1, C_2$ of $X$. ...
0
votes
0
answers
51
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How is it possible that the shortest path in a certain space approaches but never reaches a value? Is distance in this example well-defined?
Consider a space consisting of two horizontal lines at y=0 and y=1, as well as vertical lines at x=(1/n) where n is any natural number. This is the same as the deleted comb space with a top to it. ...
- 331
2
votes
3
answers
153
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Gromov-Hausdorff distance of equivalent metric spaces
Consider a set $X$ and two metrics $d_1, d_2$ on $X$ which are equivalent, i.e.
$$Ad_1\leq d_2\leq Bd_1 $$
on $X\times X$ for some positive constants $A,B$. Assume the metric space $X$ is compact.
...
- 61
2
votes
2
answers
133
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Is it possible to define a space and/or a distance function such that there is always more than 1 shortest path between any 2 points?
I am in my second semester of university in maths and physics and thought of a question I am unable to answer.
I asked my analysis teacher of the last semester if it was possible to define a space and/...
- 29
10
votes
2
answers
403
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Help proving that this metric space is not path connected
Consider the metric space embedded in $S^1$ with the intrinsic metric(the distance between two points is the length of the shortest arc connecting them):
$\hspace{3cm}$
Notice there are $3$ 'gaps' in ...
- 3,314
1
vote
0
answers
73
views
Is an open connected subspace of $\mathbb{R^2}$ locally path-connected?
I want to proof the following statement :
Let X be an open connected subspace of $\mathbb{R^2}$. Show that X is also path connected. The standard way to prove this problem from what I at least saw was ...
- 405
0
votes
0
answers
32
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Connected components and limited sets
$X \in \mathbb{R}^{n}$ is a limited subset. Prove that $\mathbb{R}^{n} - X$ has exactly one ilimited connected component, for $n > 1$.
I tried to show that if $\mathbb{R}^{n} - X$ has more than one ...
- 21
0
votes
1
answer
67
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Is this subspace of $\mathbb{R}^2$ connected? arcwise connected?
Let
$$
A \colon= \left\{ (x, y) \in \mathbb{R}^2 \colon \, 0 < x \leq 1, \ y = \sin \frac1x \right\}
$$
and
$$
B \colon= \left\{ (x, y) \in \mathbb{R}^2 \colon \, y=0, \ -1 \leq x \leq 0 \right\},
$...
- 27.2k
2
votes
2
answers
135
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Let A be a countable set of $\mathbb{R}^n , n>1$. We claim that $\mathbb{R}^n-A$ is connected.
The proof goes as follows... "Given $x,y \in \mathbb{R}^n-A,$ Choose a point $z \in [x,y]$ other than $x$ and $y$, where $[x,y]=\{a \in \mathbb{R}^n :a=(1-t)x+ty,0\leq t\leq1\}$, Now choose $w \...
4
votes
0
answers
144
views
Metric space arc-connectedness
I need hints to prove the implication “path-connected implies arc-connected” for metric spaces? [from Pugh chapter $2$ exercise $75$]
For simplicity, let me just call, given a path $f : [0,1] \to M$, ...
0
votes
1
answer
57
views
Points of boundary of open path-connected set which don't break path-connectedness
Let $X$ be a path-connected metric space, and $U\subseteq X$ an open, non-empty, proper path-connected subspace.
There can exist points $x\in \partial U$ such that $U\cup \{x\}$ is not path-connected, ...
- 11.9k
1
vote
1
answer
43
views
Connectedness of the adherent values of a sequence
Let $(X,d)$ be a compact metric space and $(x_n)$ a sequence in $X$ such that $d(x_n, x_{n+1}) \to 0$ when $n \to \infty$.
The set $\Gamma$ of the adherent values of $(x_n)$ is connected. Here is a ...
- 398
2
votes
1
answer
56
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Existence of $M>0$ s.t. every $x,y\in K\subset U$ can be connected by a path $l$ s.t. $|l|<M|x-y|$ for compact & connect $K$ and bounded open $U$?
Inspired by the brilliant answer by Martin R: https://math.stackexchange.com/a/4679552/820472 to the question Does there exist $M>0$ such that every $x,y\in U$ can be connected by a path $l$ with $|...
- 1,050
1
vote
1
answer
29
views
Does there exist $M>0$ such that every $x,y\in U$ can be connected by a path $l$ with $|l|< M|x-y|$ in an open, bounded and path-connect set $U$?
Let $U\subset\mathbb{R}^n$ be a bounded, open and path-connected set. Then every two $x,y\in U$ can be connected to each other by a polygonal chain $l$. I am wondering whether you could find an $M>...
- 1,050
2
votes
0
answers
53
views
Given this definition of geodesic, does it have constant speed ? $f(r,t)=f(r,s)+f(s,t)$
Let $(X,d)$ be a metric space, and $x,y\in X$. In Probability Measures on Metric Spaces of Nonpositive Curvature, Sturm defines a geodesic joining $x$ and $y$ as some continuous path $\gamma :[a,b]\to ...
- 657
0
votes
1
answer
93
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 ...
- 852
1
vote
1
answer
337
views
Can the notions of "path" and "connectedness" be generalized to discrete spaces?
A path in a topological space X is defined as a continuous function f from the closed interval [0,1] into X. A connected space X is defined as a space that cannot be represented as the union of two or ...
- 41
1
vote
1
answer
114
views
Prove $D(a;R_1,R_2)$ is a connected set
I was solving problems from the start of my Complex Analysis course, and I found this one (the beggining of my course focuses a lot in topology):
Prove that $D(a; R_1,R_2)$ is a connected set.
The ...
- 3,609
1
vote
2
answers
102
views
Prove that $\Bbb{R^3}\setminus\Bbb{Q^3}$ is path connected in the euclidian metric space $\Bbb{E^3}$.
I understand the concept of one coordinate moving while the rest don't change, however I can't make up the exact mapping that would prove this. Can anyone give me the concrete mapping?
- 43
1
vote
2
answers
387
views
Every open connected set $S \subset \mathbb{R}^n$ is path-connected.
$\mathbf{Attempt:}$Consider the open cover $\bigcup_{x\in S} B(x,r_x)$ of $S$ with each $B(x,r_x)\subset S$ for some $r_x>0$.
We can invoke the Lindelöf property to get ourselves a countable ...
- 2,630
0
votes
0
answers
27
views
Possibly differing definitions of local path-connectedness
In C.C.Pugh's book "Real Mathematical Analysis" a metric space $X$ is said to be locally path-connected if for each $p\in X$ and (open) neighborhood $U$ containing $p$ there is a path-...
- 124
0
votes
1
answer
84
views
Verify and understand Proof of Path connected implies connected
I was “reading” Pugh’s Mathematical Analysis Chapter 2. There, he defines a metric space to be path connected if there exists a continuous function $f : [a,b] \mapsto M$ for all points $(p,q)$ such ...
0
votes
1
answer
67
views
Let $R > r > 0$ and $A = \{ (x,y,z): r^2 \leq x^2+y^2+z^2 \leq R^2\}$ show that $A$ is path-connected
Let $R > r > 0$ and $A = \{(x,y,z): r^2 \leq x^2+y^2+z^2 \leq R^2\}$ show that $A$ is a path-connected.
A path from a point $x$ to a point $y$ in a topological space $X$ is a continuous function ...
0
votes
1
answer
371
views
In a normed vector space,open connected set is path-connected.(Proof using equivalence relation)
Suppose $(V,\Vert.\Vert)$ is a normed linear space(NLS) and $E$ is an open connected set in $V$,show that $E$ is pathwise connected.
Proof:
Definition:A subset $E$ of a normed linear space $V$ is ...
- 3,801
0
votes
2
answers
238
views
A double cone in $\mathbb{R}^3$ is path connected.
I want to show that the following subset of $\mathbb{R}^3$ is path connected. Define a double cone as below
$$M=\{(x,y,z)\in\mathbb{R}^3\mid x^2+y^2=z^2\}.$$
The only thing I know is the definition ...
- 15.2k
-1
votes
2
answers
70
views
Show a subset is not path connected.
Show that the subset $\{(x, y) \in \mathbb R^2 \mid x\neq 0\}$ of $\mathbb R^2$ is not path-connected.
I know that $X$ is path connected if any two points in $X$ are connected by a path in $X$, but ...
- 71
0
votes
0
answers
72
views
Why $\text{GL}_2(\mathbb{C})$ is connected? [duplicate]
Why $\text{GL}_2(\mathbb{C})$ is connected?
I know this question answered several times here, but I don't want to jump directly into a rigorous proof, rather I am looking for an intuitive view.
user694028
1
vote
0
answers
32
views
Proof space is Connected Space [duplicate]
Let $A:=\{(0,y):y\in[-1,1]\}$ and $B:=\{(x, sin({1\over x}):x \in (0,1]\}$, finally, let $C:=A\cup B$
Show C is a connected space
Show C is not path-connected
1 has me stumped, I have only ever ...
- 2,289
1
vote
1
answer
359
views
Is $\overline{B_1((1,0))}\cup B_1((-1,0))$ connected? Is it path connected?
I'm trying to solve a question which asks me to let $B_1(p)$ denote the unit ball around $p$ in $\mathbb{R}^2$. I'm supposed to decide whether $\overline{B_1((1,0))}\cup B_1((-1,0))$ is connected and/...
- 11
0
votes
0
answers
310
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 ...
- 981
2
votes
2
answers
388
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-...
- 607
5
votes
2
answers
749
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 ...
- 309
3
votes
1
answer
369
views
Do locally path connected locally compact metric spaces have an intrinsic metric?
Given a metric space $(X,d)$ we define the path metric $\rho(x,y)$ as the infimum of all paths lengths from $x$ to $y$. The length of a path $\gamma$ means the supremum of all $\sum_i d \big (\gamma(...
- 10.5k
0
votes
1
answer
145
views
Prove that a length-minimizing path in a metric space is injective
Suppose $(X,d)$ is a metric space with the nearest point property and $a,b \in X$ with $a \ne b$. Suppose there is a path of finite length in $X$ from $a$ to $b$ and let $m$ be the infimum of the ...
- 9,174
0
votes
1
answer
40
views
Example of infinite , compact, path connected metric space $X$ which is homeomorphic to $X \times X$?
Does there exist a compact, path connected metric space $X$, with more than one point , such that $X \times X$ is homeomorphic with $X$ ?
user495643
0
votes
0
answers
108
views
Prove that there exists an another path $g$ with the same image as $f$ but length of $g = tL \forall t\in [0,1]$ where $L$ is the length of $f$
Theorem: Suppose $(X,d)$ is a metric space and $f:[0,1] \rightarrow X$ is a path in $X$ with no-zero finite length $L$. Then, there exists a path $g:[0,1] \rightarrow X$ from $f(0)$ to $f(1)$ that has ...
- 9,174
1
vote
1
answer
143
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$A,B$ subsets of euclidean space, $A$ convex, $B$ path connected and closure of $A$ has a point in common with $B$
Let $A$ be a convex subset of $\mathbb R^n$ and $B$ be a path connected subset of $\mathbb R^n$ such that $\bar A \cap B$ is non empty ; then is it true that $A\cup B$ is path connected ?
Since $\bar ...
user495643
2
votes
1
answer
1k
views
function with path connected graph
Let $f : (a,b) \to \mathbb R$ be a function with path connected graph, then is $f$ necessarily continuous ?
If the domain was instead of the form $[a,b]$, then I can show $f$ is continuous from path ...
user495643
3
votes
1
answer
181
views
Every Subcontinuum of a $1$ Dimensional Locally Path Connected Plane Continuum is Locally Path Connected
EDIT The question was originally stated in a general setting, but now it is in terms of plane continua as these are the cases I care about.
A $continuum$ is a compact, connected metric space.
By a $...
- 8,336
2
votes
1
answer
312
views
Pathological Continua which are Path Connected and Locally Path Connected.
I'm doing research in generalised inverse limits, and I'm trying to prove a result about circle-like plane continua.
Definitions
A continuum is a compact, connected metric space.
A plane continuum ...
- 8,336
4
votes
4
answers
363
views
$W$ be an $m$-dimensional linear subspace of $\mathbb R^n$ such that $m\le n-2$ , then is it true that $\mathbb R^n \setminus W$ is connected?
Let $W$ be an $m$-dimensional linear subspace of $\mathbb R^n$ such that $m\le n-2$ , then is it true that $\mathbb R^n \setminus W$ is connected (hence path-connected as it is open in $\mathbb R^n$ ) ...
- 8,427
1
vote
1
answer
45
views
A property regarding open domain in complex plane
Let $V$ be an open connected subset of $\mathbb R^2$ , then is it true that for every compact set $K \subseteq V$ , there exist a compact set $A$ and an open connected set $B \subseteq \mathbb R^2$ ...
user228168
1
vote
2
answers
808
views
Union of path connected pairwise not disjoint subsets
Problem
Let $(X,d)$ be a metric space and let $\mathcal A$ be a family of path connected subsets of $X$ such that for every pair of sets $A,B \in \mathcal A$ there are $A_0,...,A_n \in \mathcal A$ ...
- 2,661
2
votes
1
answer
383
views
Showing this set is not path connected
Show that the region $\{(x,y) \in \mathbb{R}^2 : x< 0 \ \text{or} \ x>1 \}$ is not path connected.
Suppose that $\{(x,y) \in \mathbb{R}^2 : x< 0 \ \text{or} \ x>1 \}$ is path connected. ...
user319128
0
votes
1
answer
655
views
Is the unit sphere in $\Bbb R^4$ path connected?
I am asked whether the unit sphere
$$X=\{(x,y,z,w)|x^2 + y^2 + z^2 + w^2 = 1 \}\subset \mathbb{R}^4,$$
is path connected or not.
I just know that $X$ is a closed subset. How can I answer this question?...
- 3,503
2
votes
1
answer
56
views
$X \subseteq M(n,\mathbb C) ; |X|>1 ; $ connected/path connected, what about $S:=\{x \in \mathbb C : x $ is an eigenvalue of some matrix in $X\}$?
Let $X \subseteq M(n,\mathbb C)$ be a set with more than one element and $S:=\{x \in \mathbb C : x $ is an eigenvalue of some matrix in $X\}$. I know that if $X$ is compact then so is $S$. My question ...
user228168
4
votes
1
answer
470
views
how can I prove formally that this set is not path connected
Let $ \mathbb{R}^2 $ with its usual topology, let $D$ the set of all the lines that pass through the origin, with rational slope. And add to $D$ some point that does not lie in any of the lines ( call ...
- 3,113