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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.

A topological space $X$ is path-connected (or pathwise connected) if, for any $a, b \in X$, there exists a path from $a$ to $b$. That is, if there exists a continuous mapping $f:[0,1]\rightarrow X$ such that $f(0)=a$ and $f(1)=b$.

This is closely related to arc-connected spaces, in which there is an arc between any two points. That is, for any $a, b \in X$, there is a path from $a$ to $b$ which is homeomorphic to the unit closed interval.

Every arc-connected space is path-connected, and every path-connected space is connected.

For more general notions of connectedness, use the tag.

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Path-connected and locally connected space that is not locally path-connected

I'm trying to classify the various topological concepts about connectedness. According to 3 assertions ((Locally) path-connectedness implies (locally) connectedness. Connectedness together with locally path-connectedness implies…
Qian
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Topologist's sine curve is not path-connected

Is there a (preferably elementary) proof that the graph of the (discontinuous) function $y$ defined on $[0,1)$ by $$ y(x) =\begin{cases} \sin\left(\dfrac{1}{x}\right) & \mbox{if $0\lt x \lt 1$,}\\\ 0 & \mbox{if $x=0$,}\end{cases}$$ is not path…
persononinternet
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Can path connectedness be defined without using the unit interval?

Can path connectedness be defined without using the unit interval or more generally the real numbers? I.e., do we need Dedekind cuts or Cauchy convergence equivalence classes of the rational numbers (metric space completion) in order to define any…
Chill2Macht
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Space which is connected but not path-connected

Consider the following two definitions: Connected : A topologiocal space X is connected if it is not the disjoint union of two open subsets, i.e. if X is a disjoint union of two open sets A and B, then A or B is empty set. Path Connected : A…
Chen M Ling
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Does path-connected imply simple path-connected?

Let $X$ be a path-connected topological space, i.e., for any two points $a,b\in X$ there is a continuous map $\gamma\colon[0,1]\to X$ such that $\gamma(0)=a$ and $\gamma(1)=b$. Note that beyond continuity little is required about $\gamma$. Is it…
Hagen von Eitzen
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Smooth curves on a path connected smooth manifold

Suppose that $M$ is a path connected smooth manifold, so any two points $p,q\in M$ can be joined with a continuous curve on $M$. Is it true that any two points can be joined with a smooth (I mean $C^{\infty}$) curve on $M$?
Dubious
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Arcwise connected part of $\mathbb R^2$

Here's a question that I share: Show that if $D$ is a countable subset of $\mathbb R^2$ (provided with its usual topology) then $X=\mathbb R^2 \backslash D $ is arcwise connected.
Mohamed
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Geometry of the set of coefficients such that monic polynomials have roots within unit disk

We let $\pi$ be the bijection between coefficients of the real monic polynomials to the real monic polynomials. Let $a\in \mathbb R^n$ be fixed vector. Then \begin{align*} \pi(a) = t^n + a_{n-1} t^{n-1} + \dots + a_0. \\ \end{align*} Now denote the…
user1101010
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Is the set $\{X \in \mathcal{M}({m \times n}) : \rho(M-NX) < 1\} $ connected?

Suppose $M \in \mathcal M(n \times n; \mathbb R)$ and $N \in \mathcal M(n \times m; \mathbb R)$ are fixed with $N\neq 0$. Let \begin{align*} E = \{X \in \mathcal{M}(m \times n; \mathbb R) : \rho(M-NX) < 1\}, \end{align*} where $\rho(\cdot)$…
user1101010
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Proving that the join of a path-connected space with an arbitrary space is simply-connected

I am struggling with the following question from Allen Hatcher's algebraic topology book. Define the join $X*Y$ of two topological spaces $X$ and $Y$ to be the quotient of $X\times Y \times I$ under the following identifications: $(x,y,0)\equiv…
Roy Ben-Abraham
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Connected But Not Path-Connected?

Can you think of any spaces that are connected but not path connected apart from the Topologist's Sine Curve?
rk101
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Does there exist a continuous path between two sets of oriented basis for a vector space out of a collection of subspaces?

Let $V_1, V_2, \dots, V_n$ be a collection of vector subspaces in $\mathbb R^n$. For each $j=1, \dots, n$, $\dim(V_j) = m$ with $2 \le m < n$. We also have the condition: for any collection of $\lceil{\frac n m}\rceil$ vector spaces from $\{V_1,…
user1101010
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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 scales of the graph. As can be seen, for small…
mweiss
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If $H$ and $G/H$ are path connected, then is $G$ path connected?

Let $G$ a topological group and $H$ a subgroup such that $H$ and $G/H$ are path connected. Is it true this implies that $G$ is path connected? I already know that if $H$ and $G/H$ are connected so is G (proposition 1.6.5), but i can't find anything…
Lucas Moreira
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Prob. 2(b), Sec. 25, in Munkres' TOPOLOGY, 2nd ed: The iff-condition for two points to be in the same component of $\mathbb{R}^\omega$

Here is Prob. 2 (b), Sec. 25, in the book Topology by James R. Munkres, 2nd edition: Consider $\mathbb{R}^\omega$ in the uniform topology. Show that $\mathbf{x}$ and $\mathbf{y}$ lie in the same component of $\mathbb{R}^\omega$ if and only if the…
Saaqib Mahmood
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