# 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.

666 questions
Filter by
Sorted by
Tagged with
61 views

### 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 ...
• 989
34 views

### 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....
• 87
1 vote
69 views

### 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 ...
• 137
1 vote
59 views

### 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 $\tilde X$ ...
118 views

### 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/...
51 views

• 437
64 views

### 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 ...
• 1,548
216 views

• 1,218
80 views

### 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 ...
1 vote
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 ...
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?
64 views

• 18.5k
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 ...
• 8,541
70 views

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: ...
• 411