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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.

674 questions
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Path-connectedness in the cofinite topology

I recently proved the following for a homework exercise. Let $X$ be an infinite set endowed with the cofinite topology (open, iff finite complement or empty). If the cardinality of a subset $U\subseteq X$ is at least that of $\mathbb R$ (i.e.…
Cubi73
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Erdos Number Technicalities

Imagine this: I publish a paper with my friend who doesn't yet have and Erdos number. He later gets an Erdos number of 4, or any number for that matter. Is my Erdos number automatically 5 (one greater than my friends), or do I need to publish…
mtheorylord
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how to prove that a subset is Path-connected

For example, if you are given the subset S = {a < arg(z-z_0) < b, with 0 < a < b < π} how do you show that it is path connected? I know, that I need to show that there exists a path from one arbitrary point to another arbitrary point, say a path…
Mikethemathnerd
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Map-Making Point Connectivity Problem

GOAL: Create the longest routes (in km) that follows the rules set below (for the game Need for Speed: Rivals) RULES: No road (connection between two points) may be used twice. Single points CAN be used as many times as necessary, so long as the…
Aidan Moore
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Path-Connectedness of Inverse Mapping

Let $\varphi: [a,b] \rightarrow \Bbb R^3$ be a continuous path and $a
delog
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Plasuible Argument that $A$ must be an interval where $A \in R$ be path-connected.

" Let $A\in \Bbb R$ be path-connected. Give plausible argument that $A$ must be an interval(closed/open/or half-open). Are things as simple in $R^2?$ " My textbook on analysis mentioned such as above. What is the "plausible argument" that the…
delog
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Find position on a path

I am building a 2D game, I got a path of $N$ coordinates where $N$ is a small integer. I calculated the path total length to be $L$, and I decided that I would like to complete the path within some fixed time, so during each frame, I am calculating…
Ilya Gazman
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Proof or disprove: If $C\subseteq \mathbb{R}^n$ is path connected, then the interior $C^°$ is also path connected.

Proof or disprove: If $C\subseteq \mathbb{R}^n$ is path connected, then the interior $C^°$ is also path connected. As I'm studying for my upcoming Analysis II Exam, I found an exercise in an old exam. I would really appreciate, if someone could help…
Isaac16726
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