Articles of path connected

Path-connected implies continuous?

At the beginning of proof of problem 2, The author choose arbitrary a in (0,1) and Claimed that there exists some closed interval [a1,a2] containing a. I know this is right, but I’m curious that this is always guranteed or is just specific case. Is it possible for some open sets not to have a […]

$W$ 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 $\mathbb R^n$ ) ? My motivation comes from what I can intuitively feel is that if we remove a straight line from […]

Do linear continua contain $\mathbb{R}$? Can a nontrivial connected space have only trivial path components?

Consider a connected space $X$ under the order topology, such that $X$ contains at least two elements. Does there exist a subspace of $X$ homeomorphic to $\mathbb{R}$? If there isn’t, then $X$ has only trivial path components.

Is Every (Non-Trivial) Path Connected Space Uncountable?

I know that every non-trivial metric space with more than one point which is connected is uncountable. However, if we don’t demand that the space be a metric space, we can find examples of such odd sets as countable, Hausdorff, and connected spaces. My question: given a path connected space with more than one point, […]

Is the unit sphere in $\Bbb R^4$ is path connected?

I am asked whether $X=\{(x,y,z,w)|x^2 + y^2 + z^2 + w^2 = 1 \}\subset \mathbb{R}^4$, is path connected or not. I just know that $X$ is a closed subset. How can I answer this question? Is there any hint? Thank you very much.

Graph Theory Path Problem

Let $G$ be a connected graph such that $δ(G)≥k$. Proof there exists a path $P$ of length $k$ such that $G-P$ is connected. I am not sure how do I actually approach this question. Do I consider contradiction or inductive proof, if so how. Really appreciate it alot

Topologist's sine curve is not path-connected

Is there a (preferably elementary) proof that the graph of the 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 connected?