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.
691 questions
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Situation where a non-$T_0$ path-connected space is arc-connected
Suppose in all that follows that $X$ is a path connected topological space. That is, between any two points $a$ and $b$ there is a path $f:[0,1]\to X$ joining $a$ to $b$.
It is a classical result (...
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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 ...
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Can path-connectedness be defined using connectedness?
Can there be a "useful" and a non-trivial property $P$ for general spaces such that we have "Path-Connectedness $\iff$ Connectedness $\land$ $P$"?
I know "Path-Connectedness $\...
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Showing that a statistic of a random set is measurable
I would like to describe a random set that almost surely contains the origin by supremum of magnitudes of all points in the path connected component of the random set containing the origin. In other ...
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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 ...
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If $f: U \to \mathbb{R}$ such that for every diffrentiable curve $\gamma:[0,1] \to U$, $f \circ \gamma$ is constant a.e. Is $f$ constant a.e.?
Let $U \subset \mathbb{R}^n$ be a domain (open connected set), and let $f: U \to \mathbb{R}$ be a a function such that for every differentiable path $\gamma: [0,1] \to U$ we have that $f \circ \gamma$ ...
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Help with showing $M$ is path connected if and only if $M$ is connected and locally path-connected
I'm trying to solve problem 10 of chapter 3's exercises in Elementary Classical Analysis by Jerold Marsden and Michael Hoffman. The problem asks
A metric space $M$ is said to be locally path-...
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Connectedness and Path-Connectedness of Subset of $\mathbb{R}^2$ with exactly one irrational coordinate.
Consider the set $S\subset \mathbb{R}^2$ of all points which have exactly one irrational coordinate. That is $$x=(x_1,x_2)\in S \iff x_1 \text{ or } x_2 \text{ irrational but not both irrational}$$
Is ...
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$\sigma$-connected spaces and path-connected property
A topological space $X$ is called $\sigma$-connected if it cannot be written as a union of at least two and at most countably infinitely many pairwise disjoint nonempty closed sets. In other words, $...
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Is the interlocking interval topology path connected?
Let $X=\{x\in\mathbb R:x>0,x\notin\mathbb N\}$ and define $\mathcal B=\{(0,\frac{1}{n}):n\in\mathbb N,n\ge 2\}\cup\{(0,\frac{1}{n}))\cup(n,n+1):n\in\mathbb N,n\ge 1\}$. It can be shown that $\...
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Are path-connected LOTS also locally path-connected?
According to https://topology.pi-base.org/theorems/T000522 (which cites Theorem 24.1 of https://zbmath.org/0951.54001 ) every connected linearly ordered topological space (LOTS) is also locally ...
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Difficulty understanding proof that the Topologist's Sine Curve is not path-connected.
I'm reading Hatcher's Introduction to Point-set Topology (https://pi.math.cornell.edu/~hatcher/Top/TopNotes.pdf)
The Topologist's Sine Curve is given as an example on page 20, as follows:
Example. ...
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Specific construction of loop in Union of path connected spaces
I have been trying to get my head around this question and its solution.
The (easier and more straightforward) solution using the Lebesgue Number Lemma is greatly explained here:
Prove that a space is ...
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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 $X$. ...
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Proving complement of a line in $\mathbb{R}^3$ is path connected [duplicate]
Question: Let $S$ be a line in $\mathbb{R}^3$. Then prove/disprove $\mathbb{R}^3\setminus S$ is path connected
(Note: $S$ may be any line in $\mathbb{R}^3$ not necessarily any axes! for example, $S$ ...
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Showing particular quotient space is (not) compact, connected and Hausdorff
I came across the following question, I'm unsure about some of my answers.
Let $U = \{(x, y) \in \mathbb{R}^{2} \mid y \in \{0, 1\}\}$ be a subspace of $\mathbb{R}^{2}$. We define the following ...
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Path-connected components and homotopy equivalence of topological space $\Delta$ of ordered triangles in $\mathbb{C}$.
I am having trouble with the following question, particularly the intuition behind it and visualizing the described space.
Consider the subspace $\Delta$ of $\mathbb{C}^{3}$, of ordered triangles in $...
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Differential Forms and Path Integrals
How to prove this problem from Fulton Book (Algebraic Topology)
Show that an open set U in the plane is connected if and only if there is a segmented path between any two pints of U. Can you show that ...
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Is Rudin's definition of simple connectedness in his RCA equivalent to the usual one?
At the bullet point $10.38$ in his Real and Complex Analysis, Rudin gives a very quick rundown on homotopy and simple connectedness, but the definition he gives is the following:
If [a topological ...
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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. ...
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Path connected Julia set
Is it true that if a polynomial of degree at least 2 has a connected Julia set, then this set is path connected? It would be great to get a reference in either case. Thanks a lot!
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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.
...
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Path existence between local maxima
Let $f:\mathbb{S}^2\to\mathbb{R}$ be a smooth function and let $M=\{x_1,\dots x_K\}$ be the set of its local maxima. I am trying to determine if there is a way to find out if there exists a '...
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Is the following connected space path connected?
Let us consider the topological space $(\mathbb R,\mathcal O)$, with topology $\mathcal O:=\{A\subseteq\mathbb R:A\subseteq\mathbb Q\mbox{ or } \mathbb Q\subseteq A \}$, where $\mathbb Q$ is the set ...
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Counting $10$ length paths in a $2 \times 4$ rectangle with distance $6$ units from start to end meaning negative moves allowed?
How many different routes of length 10 units (each side is 1 unit) are there to traverse from lower left corner (point A) to top right corner (point B) in a rectangle with 2 rows and 4 column cells ...
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Collapsing the line gives a quotient of quasi-circle to the circle
Trying to solve this problem $1.3.7$ from Hatcher, but the description already confuses me. I know if we didn't include the segment connecting $y = \sin(\frac{1}{x})$ and segment $ [-1,1]$ in $y$ ...
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Is it possible to fill a 2d grid with a given group of rectangles, so that every rectangle can trace a ortogonal path to a single point in the grid.
While playing a city builder game, I noticed that the area of the buildings I can place on the game map is lower than the area of the map itself, since I need to place the roads that connect all ...
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How to make sense of $\int _{\gamma }f(z)\,\left|dz\right|$?
Context
I understand from [1] that to define the contour integral, let $f:\mathbb {C} \to \mathbb {C} $ be a continuous function on the directed smooth curve $\gamma$. Then the integral along $\...
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Show that $\mathbb{R}^n\setminus K$ is connected
Let be $K$ a non-empty compact, convex subset of $\mathbb{R}^n$, where $n>1$. Show that $S:=\mathbb{R}^n\setminus K$ is connected.
As $K$ is bounded there exists a real number $c\geq 0$ such that $...
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What exactly is the proof of onto here?
I have been reading the solution of the following question:
We can regard $\pi_1(X, x_0)$ as the set of basepoint-preserving homotopy classes of maps $(S^1,s_0) \to (X, x_0).$ Let $[S^1, X]$ be the ...
user965463
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Show that unit circle is arcwise connected in $R^2$
What I have tried
Consider two arbitrary points on the unit circle, $P = (\cos \theta_1, \sin \theta_1)$ and $Q = (\cos \theta_2, \sin \theta_2)$, where $\theta_1$ and $\theta_2$ are angles in radians....
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Arcwise connected components under a retraction on two points
Let $X$ be an Hausdorff space, and let $p,q$ two distinct points. Then if $X$ retracts on $\{ p,q\}$, establish if $X$ has at least two arcwise connected components, or if $X$ has at most two arcwise ...
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Image of paths closed in a covering
Let $p:(\tilde X, \tilde x_0)\rightarrow(X, x_0)$ be a covering map that is path-connected and locally path-connected. Is it true that given $\gamma$ and $\gamma'$, two continuous paths in $\tilde X$ ...
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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/...
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Set of all Complex Paths
A complex path $\gamma$ (path at $\mathbb{C}$), can be defined like a continuous function
$$\gamma: \left[ a,b \right] \subset \mathbb{R} \to \mathbb{C}$$
$$\gamma \left( t \right) = x \left( t \right)...
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Does path connectedness of $Cl(A)$ implies $A$ is connected? [closed]
I recently came across a problem in which it was necessary to prove that $\mathbb Q^2 \cup \mathbb I^2$ is connected but and I went through the path connectivity of $Cl(\mathbb Q^2 \cup \mathbb I^2)$. ...
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Help proving that this metric space is not path connected
Consider the metric space embedded in $S^1$ with the intrinsic metric(the distance between two points is the length of the shortest arc connecting them):
$\hspace{3cm}$
Notice there are $3$ 'gaps' in ...
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Proving that $\gamma:[0,1]\to V$ is continuous
I'm trying to prove that if $(V,|\cdot|)$ is a normed vector space over $\mathbb C$ and $x,y\in V$, then the map $$\gamma:[0,1]\to V$$
defined by $\gamma(t)=x(1-t)+yt$ is continuous, where $[0,1]$ is ...
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Monotone Path: For any two points $x,y$, there exists an increasing path $\gamma$ with terminals $x,y$ and $f\circ\gamma$ is monotonic along the path.
Consider a function $f: \Bbb R^n \to \Bbb R$ and points $x, y \in \Bbb R^n$. We say that $\gamma$ is an “increasing path from $x$ to $y$” if
$\gamma: [0, 1] \to \Bbb R^n$ is continuous with $\gamma(0)...
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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 ...
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Is an immersively path-connected space immersively injectively path-connected?
Note: This is a followup to another question of mine. The statement is almost the same; the only difference is that I ask for injective path-connectedness instead of arc connectedness.
Suppose $X$ is ...
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Is an immersively path-connected space arc-connected?
Edit: I've asked a followup question.
Suppose $X$ is a topological space such that any two points $x_0,x_1\in X$ are connected by an immersive path, i.e. there is a locally homeomorphic embedding $\...
- 36.8k
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Non injectively path-connected space
Do you have an example of a path-connected non-hausdorff space on which two points can't be injectively path-connected? (that is, any path between them is not injective). I tried to figure out what ...
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Proving that the long line isn't path-connected via arcwise-connectedness, searching for easier proof
Searching on the internet I found the following result: Every Hausdorff space that is path-connected is also arcwise-connected (for every two points of the space there's a path between them that is ...
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Prove $\mathbb{C}^n \setminus X$ is path connected.
Let $f \in \mathbb{C}[z_{1}, . . . , z_{n}]$ be a nonzero polynomial ($n ≥ 1$) and
$X = \{ z ∈ \mathbb{C}^n| f(z) = 0 \}.$
How do we prove that $\mathbb{C}^n\setminus X$ is path connected?
In one ...
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Adding point to connected open set
Let $X$ be a compact, connected, locally connected space. Let $U$ be a connected open subset of $X$. Let $p\in \overline U$. Clearly $U\cup \{p\}$ is connected.
Is $U\cup \{p\}$ locally connected?
Is $...
user410185
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Why does the existence of a dispersion point imply total path disconnectedness?
In a comment of Fractal of the topologist's sine curve is connected and totally path-disconnected? M W asserts that the existence of a dispersion point, a point for which the removal of results in ...
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0
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Fractal of the topologist's sine curve is connected and totally path-disconnected?
The sin(1/x) curve is notoriously a subset of the plane which is connected but not path-connected, because of its "singularity" at the origin. I think we can make another curve which is like ...
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Alternative proof for closure of a connected set is connected
Can someone please check this proof and if it doesn’t work why? Thank you:
Let $A$ be a connected subset and $O \subset \bar{A}$ a closed and open set. Let $B = A \cap O$. Since $O$ is an open subset ...
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Zariski topology on $\mathbb{P^n}$ [duplicate]
I want to prove or disprove that the open sets of the Zariski topology on $\mathbb{P^n}$ are path-connected.
Here $\mathbb{P^n}$ is the complex projective n-space.
Thanks for any hint or answer.
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