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.

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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 ...
Brain's user avatar
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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....
Ali's user avatar
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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 ...
Dungessio's user avatar
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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 $\tilde X$ ...
Andreadel1988's user avatar
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2 answers
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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/...
Etienne8463's user avatar
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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)...
Gabriel Fanini's user avatar
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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 $Cl(\mathbb Q^2 \cup \mathbb I^2)$. ...
noname's user avatar
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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 ...
Carlyle's user avatar
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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 ...
Eduardo Magalhães's user avatar
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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)...
High GPA's user avatar
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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 ...
muhammed gunes's user avatar
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1 answer
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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 ...
tomasz's user avatar
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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 $\...
tomasz's user avatar
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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 ...
Arthur Filippi's user avatar
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Injective path in path-connected space [duplicate]

In any path-connected metric space, can we connect any two points with an injective path? This seems possible, for instance by "removing" the points where the loop is not injective, but I ...
Arthur Filippi's user avatar
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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 ...
topologicat's user avatar
3 votes
2 answers
122 views

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 ...
Ezed's user avatar
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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 $...
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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 ...
Steven Clontz's user avatar
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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 ...
Hew Wolff's user avatar
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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 ...
Kalagan's user avatar
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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.
100nanoFarad's user avatar
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Three-dimensional analog to a discrete two-dimensional spiral

Starting at some point in an infinite 2D grid, there is a simple and intuitive path to visit every other point exactly once using only unit-length movements in the directions of the basis vectors. It'...
klkj's user avatar
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3 votes
1 answer
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The complement of an algebraic set in $\mathbb{P}^n$ is path-connected

I'm studying the marvellous book "Algebraic Curves and Riemann Surfaces" of Rick Miranda and I've found this problem (III.5.Q pag.103). Actually it's not related to Riemann surfaces, but I ...
100nanoFarad's user avatar
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28 views

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 ...
ganom51's user avatar
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Set of coefficients of degree three monic real polynomial with three real roots is connected.

Let $p(x)=x^3+ax^2+bx+c$ be a cubic polynomial with real coefficients $a, b, c,$ and define: $$D=\{(a,b,c)\in \mathbb{R}^3\mid \text{the polynomial}\ p(x)\ \text{factors into linear factors over }\ \...
nkh99's user avatar
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For a continuous injective $f:[0,1)\to\mathbb{R}^2$ with no limit at $t=1$, is the complement of the image of $f$ path connected?

Suppose that $f:[0,1)\to\mathbb{R}^2$ is a continuous injection, and that $\displaystyle\lim_{t\to1^-}f(t)$ does not exist (and is not $\infty$). Is it true that $\mathbb{R}^2\setminus f([0,1))$ is ...
Jianing Song's user avatar
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4 votes
2 answers
216 views

Dugundji: cofinite topology on countable infinite set is totally path-disconnected

In James Dugundji's Topology (1966), chapter V, are the following problems: 1.3 Let $X$ be a[n infinite] countable set, with topology $\cal T = \{\emptyset\} \cup \{ A \space \vert \space {\...
Hew Wolff's user avatar
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1 vote
1 answer
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Algorithm for finding a Hamiltonian path in a DAG

Consider a directed acyclic graph (DAG) $G$ with $n$ vertices that does not possess a Hamiltonian path. Which algorithm can be employed to determine the minimum number of additional edges required to ...
ABB's user avatar
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Functions with equal gradient and path-connected open domain [duplicate]

Consider the following exercise: (1.1) If $f,g: \mathbb{R}^n\to\mathbb{R}$ are functions such that their partial derivatives exist (but are not necessarily differentiable) and $\nabla f(x)=\nabla g(x)...
user926356's user avatar
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3 votes
2 answers
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Equivalence relationship defined over membership of a in halo of b. (Non-Standard Analysis, Monads, Halos)

Let's say we define an equivalence relationship such that $a \sim b \iff b\in \mu(a), \: \mu(a)$ is the halo\monad of $a$. By definition this would include all points an infinitesimal distance away ...
Roman Schiffino's user avatar
1 vote
1 answer
78 views

Does path connectedness simplify the proof of the LES in reduced homology?

I'm reading Weintraub's Fundamentals of Algebraic Topology, in which there is an exercise (3.4.6 in the first edition) that wants us to show that for every path connected subspace $A$ of a path ...
Martin Frenzel's user avatar
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49 views

What are the elements of $\Pi_0$ on a topological spaces?

Since a set is disconnected if and only if there are non trivial clopen sets, are those clopen sets exactly the distinct path components?
user1218897's user avatar
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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\}, $...
Saaqib Mahmood's user avatar
8 votes
1 answer
142 views

(Path) connected components of zero-sets of limiting functions.

$\newcommand{\set}[1]{\{#1\}}$ $\newcommand{\R}{\mathbb R}$ $\newcommand{\bd}{\text{Bd}}$ Let $X$ be a compact metric space and $f_n:X\to \R$ be a continuous function on $X$, one for each $n$. Assume $...
caffeinemachine's user avatar
7 votes
0 answers
63 views

Must injective path-connected maps/“path-Darboux” functions be continuous?

(Inspired by https://mathoverflow.net/questions/235893/does-there-exist-a-bijection-of-mathbbrn-to-itself-such-that-the-forward-m?rq=1) Let us define a Darboux function or connected map to be a map ...
D.R.'s user avatar
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2 votes
1 answer
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Homotopy along a path

Hello I have the following question. Let $\gamma$ be a path in $X$ then we say that two loops $f_0,f_1$ are homotopic along $\gamma$ iff there is a continuos map $H:I^2 \to X$ s.t $H(t,0) = f_0(t), H(...
user1072285's user avatar
4 votes
1 answer
117 views

If a closed subset $C \subseteq \mathbb R^2$ is "fragilely"/“minimally” simple connected, must it be all of $\mathbb R^2$?

It is a corollary of the Riemann mapping theorem that every path-connected simply-connected open subset of $\mathbb R^2$ is homeomorphic to $\mathbb R^2$ (there are more elementary proofs as well ...
D.R.'s user avatar
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3 votes
1 answer
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Show that $X$ is path-connected if $Y$ and $X/Y$ are path-connected for $Y \subset X$.

I could use some help on this question: Suppose that $Y \subset X$ is a closed, path-connected subset of a topological space $X,$ and suppose that the quotient space $X/Y$ is path connected. Show ...
TheSenate's user avatar
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2 votes
2 answers
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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 \...
Praveen Kumaran P's user avatar
3 votes
0 answers
66 views

Connected Space with uncountably many path components [duplicate]

Here is a question I saw today: Provide an example of a connected topological space having uncountably many path components. Actually I have came up with an example which I believe should be correct: ...
Runyang Wang's user avatar
2 votes
0 answers
80 views

Quotient of R by irrationals with same absolute value

while studying some general topology I came across the following topological space: on $\mathbb{R}$ we define an equivalence relation by $x \sim y$ if and only if $x=y$ or $|x|=|y|$ and $x \notin \...
Pawel02's user avatar
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5 votes
0 answers
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Existence of paths obeying partial ordering

Consider a partially ordering on $\mathbb{R}^n$ that forms a lattice, with meet and join continuous w.r.t. the standard topology (i.e. a topological lattice). Can we choose a path $\gamma(t)$ with ...
psychicmachinist's user avatar
4 votes
0 answers
131 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 $f : [0,1] \to M$, ...
Fernandeez nuts's user avatar
1 vote
1 answer
33 views

Is the hypograph of a bounded function path connected?

Let $X=\left\{\left(x_1, x_2, ..., x_n\right) \in \mathbb{R}^n \vert x_i \geqslant 0, \forall i \mbox{ and }x_1+x_2+...+x_n=1\right\}$. Consider $f: X\rightarrow \mathbb{R}$. There is a proof in a ...
Ypbor's user avatar
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0 votes
1 answer
48 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, ...
Jakobian's user avatar
  • 9,957
3 votes
1 answer
195 views

Munkres' proof that topologist's sine curve is not path connected

I was reading Munkres' proof that topologist's sine curve is not path connected. What confused me was why he bothered to reduce the path $f:[a,c]\to\bar{S}$ to $f:[b,c] \to \bar{S}$. As is understood ...
zyy's user avatar
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1 vote
1 answer
35 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 ...
Kieran McShane's user avatar
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1 answer
68 views

Does $f(z)=\frac{e^z+\sin(z)}{z}$ have a complex antiderivative on $\mathbb{C}/\{0\}$

The way the question was asked in our complex analysis course made me think it doesn't have. I wanna use Cauchy' Theorem to show that $\int_{\gamma}f(z)dz\neq0$ but I have a hard time finding a good ...
MilesDefis's user avatar
0 votes
2 answers
55 views

Existence of holomorphic Antiderivative of$f(z)=\frac{z^2}{z^2-1}, \quad f: \{z\in \mathbb{C}: |z| >2\} \to \mathbb{C}$

I am supposed to show that for $f(z)=\frac{z^2}{z^2-1}, \quad f: \mathbb{C}\{-1,1\} \to \mathbb{C}$ the restriction of f on $U := \{z\in \mathbb{C}: |z| >2\} $ has a holomorphic Antiderivative. ...
MilesDefis's user avatar

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